OK, I had a good idea how to prove Aug = Sh in all dimensions, at least for any front diagram which admits a pixellation (maybe this means any front diagram after an appropriate isotopy), without much mucking about.

Recall that “pixellation” just means that we draw our front diagram as a grid diagram; that is the Hopf link looks like this hhl_pixelated.  See section 3.5 of [STZ] for more details.  The point is that, in the sheaf world, any object which can ever have singular support in a pixellated knot at infinity is an iterated cone of standard/costandard square unknots.  In particular, it follows formally that any sheaf object can be reconstructed by iteratively taking Homs from unknots. 

Now the point is, with the “global” version of the augmentation category in hand, to show Aug = Sh, we just have to match this structure on the Aug side.

I think what I am saying would be expressed by Seidel and co as: we should prove that “the unknot generates the augmentation category”.  (It would be necessary first to make the augmentation category something for which the above sentence makes sense; possibly it already makes sense without further thought but I am not entirely sure.  In particular, maybe it is necessary to make sense of the augmentation category for Legendrian graphs, but I think this is probably not any real work.)  We already know the analogous statement for sheaves, so then matching unknots, we are done.