Minimum-volume asymptotics for Boltzmann random triangulations of the d-sphere

Let $\mathcal{S}^{(d)}_n$ be the collection of triangulations of the $d$-sphere, and let $\mu^{(d)}_{n,\beta}$ denote the median of the minimum volume of the Boltzmann random sphere $R^{(d)}_{\beta,n}$ at inverse temperature $\beta\in[0,\infty)$. Boltzmann minimum-volume asymptotic conjecture. For every $d\geq 2$ and $\beta\in[0,\infty)$, there are constants $e^{(d)}(\beta),c_{d,\beta}>0$ such that

$$\mu^{(d)}_{n,\beta}=c_{d,\beta}n^{e^{(d)}(\beta)}+o\bigl(n^{e^{(d)}(\beta)}\bigr).$$

This extends the conjectured sharp upper-bound exponent for uniformly random triangulations to the more general Boltzmann model. The corresponding statement is presented as an adaptation of the uniform-model conjecture; the supplied context does not state which cases are known or whether this formulation has been resolved.