The limit shape conjecture for minimal membranes

Let $L$ be a unimodal loop, let $tL$ denote its dilation by a positive integer $t$, and let $\mathrm{Memb}(tL)$ be the set of minimal membranes with boundary loop $tL$. For a membrane $M$, write $\langle M\rangle$ for its associated surface, and write $\langle L\rangle$ for the boundary loop.

Limit shape conjecture for minimal membranes. There exists a unique surface $S\subset\mathbb R^n$ with boundary $\langle L\rangle$ such that, for every $\epsilon>0$, there is $N>0$ for which, for every $t\geq N$, the probability under the uniform distribution on $\mathrm{Memb}(tL)$ that $t^{-1}\langle M\rangle$ belongs to the $\epsilon$-neighborhood of $S$ is greater than $1-\epsilon$.

This conjecture predicts concentration of uniformly random minimal membranes around a deterministic limit surface under dilation of the boundary loop. The supplied text gives no resolution status.