The SPR-neighbor OLA distance bound for phylogenetic trees

Let $T\in\widehat{\mathcal{RBT}}_n$ be a fixed phylogenetic tree, and let $T'$ be a random tree differing from $T$ by a single subtree-prune-and-regraft (SPR) move, chosen uniformly. Write $\operatorname{ht}(T)$ for the number of edges from the root to the farthest leaf, and let $d_{\operatorname{OLA}}$ denote the ordered leaf attachment (OLA) distance. SPR-neighbor OLA distance conjecture. The expectation satisfies the asymptotic upper bound

$$\mathbb E\bigl(d_{\operatorname{OLA}}(T,T')\bigr)=O\bigl(\operatorname{ht}(T)\bigr).$$

The conjecture is supported by a computational experiment sampling SPR neighbors; the paper also describes a heuristic suggesting a bound by $8$ times the tree height. The source does not state that the conjecture has been resolved.