Tolerance complexes of collapsible complexes are Leray

Let $K$ be a $d$-collapsible simplicial complex, and let $t\geq 0$. Define the $t$-tolerance complex of $K$ by

$$\mathcal{T}_{t}(K)=\{\eta\cup\tau:\eta\in K,\ \tau\subset V,\ |\tau|\leq t\}.$$

Collapsible-to-Leray tolerance conjecture. Then $\mathcal{T}_{t}(K)$ is $(\eta(d+1,t+1)-1)$-Leray.

This is stated as a weaker version of the conjectures asserting preservation of collapsibility and Lerayness under the same tolerance operation. The source gives no resolution of this conjecture, although it records special cases, including the $d=2$, $t=1$ case.