Limiting domination number conjecture for proximity catch digraphs on simplices

Let $\mathcal{X}_n$ be a set of iid random variables uniformly distributed on a simplex in $\mathbb{R}^d$. For $r\geq 1$, let $M$ be a point in the simplex (or in the indicated region $\mathscr P_r$), let $\gamma_n(r,M,d)$ denote the domination number of the proximity catch digraph based on the extended proximity region $N_{PE}^r(\cdot,M)$, and let $p_{r,d}$ be the parameter appearing in the limiting Bernoulli distribution. Write $t_1(r),\ldots,t_{d+1}(r)$ for the specified points of the simplex, and let $\mathcal{S}(\mathcal{Y})^o$ denote the interior of the simplex.

Limiting domination number conjecture. As $n\to\infty$, the domination number satisfies

\gamma_n(r,M,d)\stackrel{\mathcal L}{\longrightarrow}\left\lbrace\begin{array}{ll} d+\operatorname{BER}(1-p_{r,d})&\text{for $r\in[1,(d+1)/d)$ and $M\in\{t_1(r),t_2(r),\ldots,t_{d+1}(r)\}$,}\\ \leq d-1&\text{for $r>(d+1)/d$ and $M\in\mathcal{S}(\mathcal{Y})^o$,}\\ d+1&\text{for $r\in[1,(d+1)/d)$ and $M\in\mathscr P_r\setminus\{t_1(r),t_2(r),\ldots,t_{d+1}(r)\}.$end{array}\right.

This conjecture describes the asymptotic domination number in the different parameter regimes for proximity catch digraphs on a simplex. The notation for $p_{r,d}$ and the geometric regions is inherited from the surrounding construction; the parser supplies no evidence that the conjecture has been resolved.