The very colorful Helly conjecture for d-Leray complexes and matroids

Let $d\beq 1$, $r\geq d+1$, $1\leq m\leq r$ and $m\leq k\leq \min\{m+d,r\}$ be integers. Let $V$ be a finite set with $|V|\geq \max\{m+d,r\}$. Let $X$ be a $d$-Leray simplicial complex on vertex set $V$, and let $M$ be a matroid of rank $r$ on vertex set $V$ with rank function $\rho$. Assume that for every $U=\{u_1,\ldots,u_d,v_1,\ldots,v_m\}\subset V$ with $\rho(U)\geq k$, there is some $i\in[m]$ such that $\{u_1,\ldots,u_d,v_i\}\in X$. Then, there is some $\tau\in X$ such that $\rho(V\setminus \tau)\leq k-1$. Very colorful Helly conjecture. This conjectural strengthening of the very colorful Helly theorem for $d$-Leray complexes would extend the theorem by incorporating a matroid-rank conclusion; its status is not established in the supplied source.