Tomahawk JSCookMenu does not trigger any actions after upgrading to JSF2 - jsf-2

Question

I've just migrated some medium-size web application from JSF 1.2/MyFaces to JSF 2.0/Mojarra. Apart from some custom components that I've implemented, everything seems to work, except for the JSCookMenu from Tomahawk (yes, I have Tomahawk which is a library from Apache, and still want to use Mojarra - but I understood there shouldn't be any big problem). I've replaced the myfaces-api.jar and myfaces-impl.jar with the corresponding jsf-api.jar and jsf-impl.jar, and took out from web.xml as many MyFaces parameters and filters as possible (some needed to remain because I still use Tomahawk). I have also upgraded Tomahawk to 1.1.11 for JSF2.
The application's JSCookMenu renders just fine, but doesn't trigger any action when clicking the menu items. I have some t:navigationMenuItem whose action attribute is bound to an outcome in faces-config.xml which should load a new view id, and yet some t:navigationMenuItem whose actionListener attribute is bound to a bean method (the Logout menu item, for example). Neither of them works, the actionListener code from my bean is not even called.
I'm not even sure how to investigate this, did anybody have a clue as to how can this be solved? I'm hoping something in web.xml can be of help.
The navigation cases are written correctly in the faces-config.xml, they used to work on JSF 1.2/MyFaces and also on JSF 2/MyFaces prior to the migration to Mojarra.
This is the relevant part of my web.xml file:
<servlet>
<servlet-name>faces</servlet-name>
<servlet-class>javax.faces.webapp.FacesServlet</servlet-class>
<load-on-startup>1</load-on-startup>
</servlet>
<servlet-mapping>
<servlet-name>faces</servlet-name>
<url-pattern>*.xhtml</url-pattern>
</servlet-mapping>
<context-param>
<param-name>com.sun.faces.allowTextChildren</param-name>
<param-value>true</param-value>
</context-param>
<context-param>
<param-name>javax.faces.FACELETS_RESOURCE_RESOLVER</param-name>
<param-value>com.avalanche.jsf.MyResourceResolver</param-value>
</context-param>
<context-param>
<param-name>javax.faces.STATE_SAVING_METHOD</param-name>
<param-value>client</param-value>
</context-param>
<context-param>
<param-name>org.apache.myfaces.AUTO_SCROLL</param-name>
<param-value>true</param-value>
</context-param>
<context-param>
<param-name>facelets.DEVELOPMENT</param-name>
<param-value>true</param-value>
</context-param>
<context-param>
<param-name>facelets.RECREATE_VALUE_EXPRESSION_ON_BUILD_BEFORE_RESTORE</param-name>
<param-value>false</param-value>
</context-param>
<filter>
<filter-name>facesExtensionsFilter</filter-name>
<filter-class>org.apache.myfaces.webapp.filter.ExtensionsFilter</filter-class>
<init-param>
<param-name>uploadMaxFileSize</param-name>
<param-value>1g</param-value>
</init-param>
</filter>
<filter-mapping>
<filter-name>facesExtensionsFilter</filter-name>
<servlet-name>faces</servlet-name>
</filter-mapping>
<filter-mapping>
<filter-name>facesExtensionsFilter</filter-name>
<url-pattern>/faces/myFacesExtensionResource/*</url-pattern>
</filter-mapping>