Agregué un filtro CORS a mi aplicación, pero parece que no se ejecuta (sin impresiones).

El filtro es este:

@Component
public class SimpleCORSFilter extends OncePerRequestFilter {
    @Override
    protected void doFilterInternal(HttpServletRequest req,
            HttpServletResponse res, FilterChain filterChain)
            throws ServletException, IOException {
        System.out.println("EXECUTING");
        HttpServletResponse response = (HttpServletResponse) res;
        response.setHeader("Access-Control-Allow-Origin", "*");
        response.setHeader("Access-Control-Allow-Methods",
                "POST, GET, OPTIONS, DELETE");
        response.setHeader("Access-Control-Max-Age", "3600");
        response.setHeader("Access-Control-Allow-Headers",
                "X-Requested-With,Content-Type");

        filterChain.doFilter(req, res);
    }
}

Luego agregué el filtro en web.xml, que es el siguiente:

<?xml version="1.0" encoding="UTF-8"?>
<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
    xmlns="http://java.sun.com/xml/ns/javaee"
    xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd"
    version="2.5">

    <!-- The definition of the Root Spring Container shared by all Servlets 
        and Filters -->
    <context-param>
        <param-name>contextConfigLocation</param-name>
        <param-value>/WEB-INF/spring/servlet-context.xml</param-value>
    </context-param>

    <listener>
        <listener-class>org.springframework.web.context.ContextLoaderListener</listener-class>
    </listener>
    <servlet>
        <servlet-name>rest</servlet-name>
        <servlet-class>org.springframework.web.servlet.DispatcherServlet</servlet-class>
        <load-on-startup>1</load-on-startup>
    </servlet>
    <servlet-mapping>
        <servlet-name>rest</servlet-name>
        <url-pattern>/*</url-pattern>
    </servlet-mapping>

    <filter>
        <filter-name>simpleCORSFilter</filter-name>
        <filter-class>org.lh.gestorepazienti.web.SimpleCORSFilter</filter-class>
    </filter>
    <filter-mapping>
        <filter-name>simpleCORSFilter</filter-name>
        <servlet-name>rest</servlet-name>
    </filter-mapping>

    <filter>
        <filter-name>OpenSessionInViewFilter</filter-name>
        <filter-class>org.springframework.orm.jpa.support.OpenEntityManagerInViewFilter</filter-class>
    </filter>
    <filter-mapping>
        <filter-name>OpenSessionInViewFilter</filter-name>
        <url-pattern>/*</url-pattern>
    </filter-mapping>
</web-app>

¿Dónde podría estar mal?