Giving keyboard focus to a GtkMenu

Question

I'm popping-up a GtkMenu at random coordinates on the screen (outside of my application's main window). I'd like to give keyboard focus to this GtkMenu.

Since I'm using Windows, I tried:

SetForegroundWindow(GDK_WINDOW_HWND(menu->window));

But that causes the menu to immediately disappear, as does:

SetFocus(GDK_WINDOW_HWND(menu->window));

I also tried:

GTK_WIDGET_SET_FLAGS(menu, GTK_CAN_FOCUS);
gtk_widget_grab_focus(menu);

But that doesn't do anything at all. Any ideas? I'm trying to make it so that the user can navigate the menu using their keyboard.