Tolerance complexes preserve Lerayness

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

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

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

This would strengthen the known tolerant Helly theorem by replacing the Helly-number hypothesis with the stronger Leray-number hypothesis. The source presents it as a conjecture; its resolution is not specified here.