The colorful fractional Helly conjecture for d-Leray complexes

A simplicial complex $\mathsf{K}$ is $d$-Leray if, for every induced subcomplex $\mathsf{L}\leq\mathsf{K}$ and every $i\ge d$, its reduced homology group $\widetilde H_i(\mathsf{L})$ over $\mathbb{Q}$ vanishes. Let the vertex set be partitioned as $N=N_1\sqcup\cdots\sqcup N_{d+1}$, and write $n_i:=|N_i|$ for $i\in[d+1]$. A colorful $d$-face is a $d$-dimensional face containing one vertex from each $N_i$. The colorful fractional Helly conjecture for $d$-Leray complexes. If $\mathsf{K}$ contains at least $\alpha n_1\cdots n_{d+1}$ colorful $d$-faces for some $\alpha\in(0,1]$, then there is an $i\in[d+1]$ such that

$$\dim\mathsf{K}[N_i]\geq\bigl(1-(1-\alpha)^{1/(d+1)}\bigr)n_i-1.$$

This would extend the proved optimal theorem for $d$-collapsible complexes to the broader class of $d$-Leray complexes, providing a topological analogue of the convex-set result; the source does not report a resolution.