Let's keep on thinking about disk with three stops.
There are a few things in common with disk with two stops and one feature in the middle, like a hole, a puncture. One need to decide if the T-branes are on the left or on the right. In the case of punctures, or several punctures, we are looking at how the standard flag in $\C^k$ map to the target flag $0 \In \C$.
I really don't know where is that target space come from?