Unmarked-boundary-loop diameter growth conjecture for modular flip-graphs

Let $\Sigma$ be a surface with unmarked and marked boundary loops, and let $\Sigma_n$ denote the corresponding surface in the family with $n$ marked points. For a surface with $k_\varepsilon$ unmarked boundary non-privileged loops, write $\operatorname{diam}({\mathcal M \mathcal F}(\Sigma_n))$ for the diameter of its modular flip-graph.

Unmarked-loop diameter growth conjecture. For every $\varepsilon>0$, there exists $k_\varepsilon$ such that, if $\Sigma$ is a surface with $k_\varepsilon$ unmarked boundary non-privileged loops, then

$$\lim_{n\to\infty}\frac{\operatorname{diam}({\mathcal M \mathcal F}(\Sigma_n))}{n}\geq 3-\varepsilon.$$

This is the unmarked-boundary analogue of the preceding conjecture and predicts a lower asymptotic diameter growth rate after sufficiently many such loops are added. The source provides no resolution, so the conjecture remains open.