The d-Matoušek-to-d-Leray conjecture

Let $d$ be a nonnegative integer, and let $\mathsf{K}$ be a simplicial complex. A simplicial complex is $d$-Matoušek and $d$-Leray as defined in the source paper. The d-Matoušek-to-d-Leray conjecture. Every $d$-Matoušek complex is $d$-Leray. The conjecture proposes that the paper's topological obstruction to $d$-representability also rules out the homological obstructions measured by the $d$-Leray property; the authors state that a proof has yet to be found.