Bounded barcode-chain conjecture for merge tree interleaving distance

Let $\mathrm{MT}_n$ be the space of merge trees with at most $n$ leaves. For merge trees $x_0,\ldots,x_k$, let $d_B(x_{i-1},x_i)$ denote their bottleneck distance, and let $d_I$ denote interleaving distance.

Bounded barcode-chain conjecture. There exists a number $M(n)$ such that for any $y_0,y_1\in\mathrm{MT}_n$, there exist $x_0,\ldots,x_k$ with $k<M(n)$, $y_0=x_0$, and $y_1=x_1$ satisfying

$$d_I=\sum_{i=1}^k d_B(x_{i-1},x-i).$$

The statement appears as a proposed consequence of the existence of a bounded-complexity path through merge-tree spaces. As written, the displayed equality does not specify the pair of endpoints to which $d_I$ refers, and the term $x-i$ appears in the source; these notational issues should be checked against the paper before relying on the formulation.