We had a long detour into Cautis-Kamnitzer's constructions, at least we had some familiarity with flag variety.
But, I still do not know about disk with 3 or more stops.
Disk with 2 stops and $k$-strands, is assigned to $D-mod(pt/GL(k) )$ which is derived equivalent to $NH_k-Mod$.
Disk with 2 stops, goes to the monoidal category $U_-$.
Disk with 3 stops with $k$ strands, we have $k+1$ many objects serving as generators, and they form an full exceptional collection.