The excellent-set conjecture for Serre conditions of simplicial complexes

Let $\Delta$ be a simplicial complex, and let $A\subseteq\Delta$ be excellent if its elements are pairwise independent and every facet of $\Delta$ contains a unique element of $A$. Define

$$\Delta_A:=\Delta-\{\sigma\in\Delta\mid \sigma\supseteq\tau\text{ for some }\tau\in A\}.$$

A simplicial complex $\Delta$ satisfies $(S_{\ell})$ when it has the stated Serre condition.

Excellent-set conjecture. If $A\subseteq\Delta$ is excellent and $\Delta$ satisfies $(S_{\ell})$, then $\Delta_A$ satisfies $(S_{\ell})$.

This conjecture proposes that deleting all faces containing members of an excellent set preserves the Serre condition. It extends the preceding result that deleting faces containing an independent set decreases depth by at most one; its status is not resolved in the supplied source.