Realizability conjecture for potential hyperbolic posets

Let $S$ be a set of compositions of $n$ into at most $s$ parts, and let $\mathcal{L}(S)$ be the associated potential hyperbolic poset obtained by taking pairwise joins and then the upward closure of those joins. A potential hyperbolic poset is called realizable if it is the hyperbolic poset of some hyperbolic slice. Realizability conjecture. Every potential hyperbolic poset is realizable. All known combinatorial properties apply to both potential and realizable hyperbolic posets, and computational realizations agree for $s\leq n\leq 6$; realization in general remains open.