Creé una clase singleton volley para mis solicitudes. Esta es mi clase singleton

public class VolleySingleton {

    private static VolleySingleton instance;
    private RequestQueue reQueue;

    private VolleySingleton(Context context) {
        reQueue = Volley.newRequestQueue(context,new OkHttpStack(new OkHttpClient()));
    }

    public static VolleySingleton getInstance(Context context) {
        if(instance == null)
            instance = new VolleySingleton(context);
        return instance;
    }

    public RequestQueue getRequestQueue() {
        return reQueue;
    }

    public <T> void addToRequestQueue(Request<T> request) {
        request.setTag("app");
        request.setShouldCache(false);
        request.setRetryPolicy(new DefaultRetryPolicy(10000, DefaultRetryPolicy.DEFAULT_MAX_RETRIES,
                DefaultRetryPolicy.DEFAULT_BACKOFF_MULT));
        getRequestQueue().add(request);
    }
}

Y estoy usando la pila OKHTTP con mi implementación de Volley. Esta es miOKHttp.class

public class OkHttpStack implements HttpStack {

    private final OkHttpClient mClient;

    public OkHttpStack(OkHttpClient client) {
        this.mClient = client;
    }

    @Override
    public HttpResponse performRequest(Request<?> request, Map<String, String> additionalHeaders)
            throws IOException, AuthFailureError {

        OkHttpClient client = mClient.clone();
        int timeoutMs = request.getTimeoutMs();
        client.setConnectTimeout(timeoutMs, TimeUnit.MILLISECONDS);
        client.setReadTimeout(timeoutMs, TimeUnit.MILLISECONDS);
        client.setWriteTimeout(timeoutMs, TimeUnit.MILLISECONDS);

        com.squareup.okhttp.Request.Builder okHttpRequestBuilder = new com.squareup.okhttp.Request.Builder();
        okHttpRequestBuilder.url(request.getUrl());

        Map<String, String> headers = request.getHeaders();
        for (final String name : headers.keySet()) {
            okHttpRequestBuilder.addHeader(name, headers.get(name));
        }
        for (final String name : additionalHeaders.keySet()) {
            okHttpRequestBuilder.addHeader(name, additionalHeaders.get(name));
        }

        setConnectionParametersForRequest(okHttpRequestBuilder, request);

        com.squareup.okhttp.Request okHttpRequest = okHttpRequestBuilder.build();
        Call okHttpCall = client.newCall(okHttpRequest);
        Response okHttpResponse = okHttpCall.execute();

        StatusLine responseStatus = new BasicStatusLine(parseProtocol(okHttpResponse.protocol()), okHttpResponse.code(), okHttpResponse.message());
        BasicHttpResponse response = new BasicHttpResponse(responseStatus);
        response.setEntity(entityFromOkHttpResponse(okHttpResponse));

        Headers responseHeaders = okHttpResponse.headers();
        for (int i = 0, len = responseHeaders.size(); i < len; i++) {
            final String name = responseHeaders.name(i), value = responseHeaders.value(i);
            if (name != null) {
                response.addHeader(new BasicHeader(name, value));
            }
        }

        return response;
    }

    private static HttpEntity entityFromOkHttpResponse(Response r) throws IOException {
        BasicHttpEntity entity = new BasicHttpEntity();
        ResponseBody body = r.body();

        entity.setContent(body.byteStream());
        entity.setContentLength(body.contentLength());
        entity.setContentEncoding(r.header("Content-Encoding"));

        if (body.contentType() != null) {
            entity.setContentType(body.contentType().type());
        }
        return entity;
    }

    @SuppressWarnings("deprecation")
    private static void setConnectionParametersForRequest(com.squareup.okhttp.Request.Builder builder, Request<?> request)
            throws IOException, AuthFailureError {
        switch (request.getMethod()) {
            case Request.Method.DEPRECATED_GET_OR_POST:
                // Ensure backwards compatibility.  Volley assumes a request with a null body is a GET.
                byte[] postBody = request.getPostBody();
                if (postBody != null) {
                    builder.post(RequestBody.create(MediaType.parse(request.getPostBodyContentType()), postBody));
                }
                break;
            case Request.Method.GET:
                builder.get();
                break;
            case Request.Method.DELETE:
                builder.delete();
                break;
            case Request.Method.POST:
                builder.post(createRequestBody(request));
                break;
            case Request.Method.PUT:
                builder.put(createRequestBody(request));
                break;
            case Request.Method.HEAD:
                builder.head();
                break;
            case Request.Method.OPTIONS:
                builder.method("OPTIONS", null);
                break;
            case Request.Method.TRACE:
                builder.method("TRACE", null);
                break;
            case Request.Method.PATCH:
                builder.patch(createRequestBody(request));
                break;
            default:
                throw new IllegalStateException("Unknown method type.");
        }
    }

    private static ProtocolVersion parseProtocol(final Protocol p) {
        switch (p) {
            case HTTP_1_0:
                return new ProtocolVersion("HTTP", 1, 0);
            case HTTP_1_1:
                return new ProtocolVersion("HTTP", 1, 1);
            case SPDY_3:
                return new ProtocolVersion("SPDY", 3, 1);
            case HTTP_2:
                return new ProtocolVersion("HTTP", 2, 0);
        }

        throw new IllegalAccessError("Unkwown protocol");
    }

    private static RequestBody createRequestBody(Request r) throws AuthFailureError {
        final byte[] body = r.getBody();
        if (body == null) return null;

        return RequestBody.create(MediaType.parse(r.getBodyContentType()), body);
    }
}

También intilicé mi Stetho en mi aplicación.

Stetho.initialize(
                Stetho.newInitializerBuilder(this)
                        .enableDumpapp(
                                Stetho.defaultDumperPluginsProvider(this))
                        .enableWebKitInspector(
                                Stetho.defaultInspectorModulesProvider(this))
                        .build());

El problema es que cuando depuro no puedo obtener las llamadas de red realizadas por mi aplicación en mi navegador. ¡No estoy seguro de dónde me estoy equivocando! ¡Ayuda!