. The interior of the corresponding boundary stratum of $C_{n,m}^+$, resp. $C_{n,k,l}^+$, is $C_{i,B}^+\times C_{n-i,([m]\smallsetminus B)\sqcup \{\bullet\}}^+$, resp. $C_{i,B_0,B_0}^+\times C_{n-i,[k]\smallsetminus B_0,[l]\smallsetminus B_0}^+$, and corresponding local coordinates are given by $$\begin{aligned} C_{i,B}^+\times C_{n-i,([m]\smallsetminus B)\sqcup \{\bullet\}}^+