The restricted interleaving metric conjecture for merge trees

Let $\mathrm{MT}$ be the space of merge trees, and let $\mathrm{MT}_n\subseteq\mathrm{MT}$ denote the subspace of merge trees with at most $n$ leaves. For merge trees in this space, write $\widehat{(d_I)}_n$ for the intrinsic metric induced by the interleaving distance, while $\widehat{d}_I$ and $d_I$ denote the unrestricted intrinsic and interleaving distances, respectively.

Restricted interleaving metric conjecture. When $n=1,2$,

$$\widehat{(d_I)}_n=\widehat{d}_I=d_I.$$

When $n>2$, there exist merge trees $(T,f),(T',f')\in\mathrm{MT}_n$ such that

$$\widehat{(d_I)}_n\big((T,f),(T',f')\big)>d_I\big((T,f),(T',f')\big).$$

This asks whether the subspace of merge trees with at most $n$ leaves is convex in the interleaving metric. The equality is known for $n=1,2$, while for $n>2$ the conjecture predicts that the restricted intrinsic metric can be strictly larger than the ambient interleaving distance.