The Helly-type compatibility conjecture for split systems on multisets

Let $\mathfrak S$ be a split system on a multiset $\mathcal M$. The quantity $\Delta(\mathcal M)$ is the parameter appearing in the known finite characterization of compatibility for split systems on multisets.

Compatibility conjecture. $\mathfrak S$ is compatible if and only if every submultiset of $\mathfrak S$ of size at most $\Delta(\mathcal M)+2$ is compatible.

This conjecture sharpens the known bound $\max\{2\Delta(\mathcal M),\Delta(\mathcal M)+2\}$ from the preceding characterization, and concerns whether compatibility can always be detected by smaller submultisets. Its status is not resolved in the supplied source.