Bootstrap has .navs that then can be specialized with .nav-tab, .nav-pill or styled differently by leaving those off.
The new sub-menu navigation uses .nav-tab, but then removes all the nav-tab styling and replaces it. If someone tried to style the actual .nav-tabs (the ones that look like paper folder 'tabs') for use elsewhere then the styles would bleed over into these navs.
Just removing the nav-tab should work fine and pick up the base nav styles:
The moremenu CSS already adds the hover states and active underline.
And the secondary-navigation would no longer need to override them:
About half of that is just remove the styles added by nav-tab.
I'd also recommend using a seperate variable for the border-top/bottom color, rather than re-use the tab one since these are navs, but not tabs in the standard theme.