The representability bound for simplicial complexes with bounded Leray number

Let $X$ be a simplicial complex with $n$ vertices, and let $h(X)$ denote its Leray number.

Representability bound conjecture. If $h(X)\leq d$ for $d\geq 1$, then

$$\operatorname{rep}(X)\leq \left\lfloor\frac{dn}{d+1}\right\rfloor.$$

Moreover, equality $\operatorname{rep}(X)=\frac{dn}{d+1}$ holds if and only if the missing faces of $X$ consist of $\frac{n}{d+1}$ pairwise disjoint sets of size $d+1$.

This conjecture extends the sharp graph case and predicts both a universal upper bound and its extremal complexes. Its resolution would strengthen the known relationship between representability and Leray number; the general case remains open.