Convergence of uniform 231-avoiding permutations to the Brownian binary tree

Let $[n]=\{1,\ldots,n\}$, let $S_n$ be a uniform $231$-avoiding permutation of $[n]$, and define its normalized graph by

$$\operatorname{gr}(S_n):=\{(i/n,S_n(i)/n):i\in[n]\}.$$

View $\operatorname{gr}(S_n)$ as a random compact subset of $[0,1]^2$, taking values in $(\mathcal{K}([0,1]^2),d_{\operatorname{haus}})$, where $d_{\operatorname{haus}}$ is the Hausdorff metric. Conjecture. As $n\to\infty$, $\operatorname{gr}(S_n)$ converges in law to the Brownian binary tree on $(0,1)$, with respect to the Hausdorff topology on $\mathcal{K}([0,1]^2)$. The conjecture proposes a universality connection between large uniform pattern-avoiding permutations and the Brownian binary tree, motivated by the comparison with the limiting tree associated with Rémy's tree growth chain; its resolution is not indicated here.