The diameter conjecture for one-vertex random Belyi surfaces

Let $S_n^O$ be the random surface in the BM model, and let $\widetilde{S}_n^O$ denote this surface conditioned on having a single vertex. Let $\widetilde{S}_n^C$ be the compactified version of $\widetilde{S}_n^O$. The notation $\operatorname{diam}$ denotes diameter, and $\xrightarrow{(P)}$ denotes convergence in probability. The one-vertex diameter conjecture. As $n\to\infty$,

$$\frac{\operatorname{diam}(\widetilde{S}_n^C)}{\log n}\xrightarrow[n\to\infty]{(P)}1.$$

The conjecture predicts that conditioning on a single vertex removes the separated high-degree vertices responsible for the $2\log n$ diameter scale in the unconditioned model. The source gives no resolution of this conditional diameter asymptotic.