What is the disk with three stops?
Disk with two stops, $k$ strands T-brane, has endomorphism algebra $NH_k$, with $q$ grading for crossing $q^{-2}$. Correspondingly, we have $(\pi: BB \to BG)_* \C_{BB}$, whose endomorphism involves $\pi^!$. Recall that for sheaves (not coherent sheaves), $\pi^! = \pi^* [\dim_\R fiber]$. This explains why we have those negative cohomological degrees.
Now we add another stop, say at the top. We will have $(k,0), (k-1,1), \cdots, (0,k)$ different types of $T$-branes.
We have a few observations:
Our goal is to build a category living over the Higgs side space. But I don't know what is the ambient space.
One guess is that, the space is still and always is $BGL_k$, and the T-brane sheaf is still $BB \to BG$ pushing forward. But this is like after doing stop removal. How to remember?
Another way of thinking is, we don't have to use $[pt/GL_k]$, but $[(pt \to pt)/GL_k]$, and then do sheaves on it. What does it mean? A sheaf of vector spaces on a space $X$, is a functor from the open category to $dgVect$. A sheaf on a quotient stack $[X/G]$, or the category of sheaves on $[X/G]$ is the limit (equalizer in the infinite version) $$ Sh(X) \to Sh(X \times G) \to Sh(X \times G \times G) \to \cdots $$
But what is $Sh(pt \to pt)$?