Perimeter approximation conjecture for the convex hull of multiple random walks

Let $L_n$ be the perimeter of the convex hull of the multiple random walks, and let ${\mathcal G}_n$ and ${\mathcal E}^{\mathrm{f}}_n$ be the sets defined in the paper. For each relevant index $k$, let $S_0^{(k)},\ldots,S_n^{(k)}$ denote the associated walk positions, let ${\boldsymbol{\mu}}_k$ be the corresponding vector, let $I^{\mathrm{f}}_{\boldsymbol{\mu}}$ be the associated index set, and let ${\mathcal G}'_n$ be the subset of ${\mathcal G}_n$ obtained by removing from ${\mathcal E}^{\mathrm{f}}_n$ (i) all $S_0^{(k)},\ldots,S_n^{(k)}$ for which ${\boldsymbol{\mu}}_k$ lies in the relative interior of its face, and (ii) all $S_0^{(k)},\ldots,S_{n-1}^{(k)}$ for which ${\boldsymbol{\mu}}_k$ is unique among $k\in I^{\mathrm{f}}_{\boldsymbol{\mu}}$. Perimeter approximation conjecture.

$$\frac{L_n-\operatorname*{perim}\operatorname*{hull}{\mathcal G}'_n}{\sqrt{n}}\xrightarrow[n\to\infty]{\mathrm{prob.}}0.$$

This conjecture would provide a perimeter-specific approximation for the general limiting theory when the number of walks satisfies $N\geq3$, where the available $L^2$ Hausdorff approximation is generally too crude to analyze the perimeter. It is motivated by the paper's results for one and two walks and by cases with three or more walks that are already settled.