The problem may lie in the fact that there is no “wrap” for the submenus, like for the top menus. Probably because they are meant to drop down and styled vertically, not to be static features. The submenu is coded in a way that css changes always affect the li items.
Perhaps this falls under the need to tweak the javascript to make the submenu present at all times in its current state?
Is it possible that a change to header.php or in another .php file could call a “submenu wrap,” or would that, too, be in the java script?