The limiting density measure conjecture for minimal membranes

Let $L$ be a unimodular loop. For a loop $L$, let $\mu_L$ be the probability distribution on $\mathrm{cloud}(L)\subset\mathbb Z^n$ defined by the normalized incidence count of lattice points in uniformly chosen minimal membranes, and define $\mu_{L,t}(a):=\mu_{tL}(ta)$. Let $S\subset\mathbb R^n$ be a limit surface.

Limiting density measure conjecture for minimal membranes. As $t\to\infty$, the measures $\mu_{L,t}$ converge to a limit measure

$$\mu_{L,\infty}:=\lim_{t\to\infty}\mu_{L,t}$$

supported on a limit surface $S\subset\mathbb R^n$.

This conjecture proposes a limiting density for lattice points on random minimal membranes and relates its support to the limit surface arising from membrane dilation. The supplied text gives no resolution status.