It's a good place to work and study, the Joshua Tree Field Station (very cool hotel).

I am still trapped by the disk with 3 stops, no punctures.

But do you understand the disk with 2 stops, how the gluing works? one disk has $k_1$ strands; another disk has $k_2$ strands, both with $2$ stops, put them together, get $k=k_1+k_2$ strands.

It is about going from $BGL(k_1)$ and $BGL(k_2)$ to $BGL(k)$. I just know $T^{k_1}$ and $T^{k_2}$ merges to $T^{k_1+k_2}$. Let's try parabolic induction: we have $$ GL(k_1) \times GL(k_2) \gets P_{k_1, k_2} \to GL(k_1+k_2) $$