Minimum-volume scaling conjecture for random triangulations of the d-sphere

Let $\mathcal{S}^{(d)}_n$ be the set of $n$-vertex triangulations of the $d$-sphere, let $M(\mathcal{S}^{(d)}_n)$ denote their minimum volume, and let $\mu^{(d)}_n$ be the median of this random variable. Minimum-volume scaling conjecture. For every $d\geq 2$ there is a constant $c_d>0$ such that

$$\mu^{(d)}_n = c_d n^{1-1/d} + o(n^{1-1/d}).$$

The conjecture asserts that the upper bound proved in the paper gives the correct asymptotic order and leading constant form. It is open even for $d=2$; in that case the available bounds constrain $c_2$ to an explicit interval.