Parameter identification conjecture for high-genus triangulation limits

Let $\mathcal{T}_{n,g}$ be the set of rooted triangulations of genus $g$ with $n$ vertices, and let $T_{n,g}$ be uniformly distributed in $\mathcal{T}_{n,g}$. For $\kappa\in(0,\frac{2}{27}]$, define

$$f(\kappa)=E\left[\left(\operatorname{degree\ of\ the\ origin\ in\ }\mathbf{T}_{\kappa}\right)^{-1}\right].$$

For $\theta\geq0$, let $\kappa\in(0,\frac{2}{27}]$ satisfy $f(\kappa)=(6(1+2\theta))^{-1}$. Parameter identification conjecture.

$$T_{n,[\theta n]}\xrightarrow[n\to\infty]{(d)}\mathbf{T}_{\kappa}.$$

This sharpens the introductory conjecture by matching the finite and infinite models through the inverse degree of the root; the required strict monotonicity and the resulting convergence are not proved in the paper.