The symmetric capacity conjecture for unimodular triangulations

Let $\widetilde{f}(d)$ be the logarithm of the number of $\mathfrak{H}$-invariant unimodular triangulations of $d\cdot\Delta_2$, and define the symmetric capacity by

$$\widetilde{c}=\lim_{d\to\infty}\frac{2\widetilde{f}(d)}{d(d-1)}.$$

Symmetric capacity conjecture. The symmetric capacity satisfies

$$\widetilde{c}=1.$$

The preceding bounds imply $\frac12\leq\widetilde{c}\leq\frac32$, while numerical regression suggests the value $1$; establishing the asymptotic limit remains open.