Local geometry for Schmidt number witnesses

Suppose that $F_E$ is the face of the convex set of all $m\otimes n$ bi-partite states which consists of states with ranges contained in a subspace $E$. For generic subspaces $E$ with a specific dimension, we use the result in [Phys. Rev. A 112 (2025), 032426] to see that there exists a number $\kappa$, depending only on the dimension of $E$, such that there exist Schmidt number $\ell$ witnesses outside of $F_{E^\perp}$ if and only if $\ell\le\kappa$. In this generic case, we show in this paper that there exist Schmidt number $\ell$ witnesses for $\ell>\kappa$ around the projection states located at the center of $F_{E^\perp}$.