The sharp upper-bound conjecture for the Yokota invariant

Let $\Gamma$ be a planar $3$-connected graph, let $\overline{\Gamma}$ denote the associated maximal-volume proper generalized hyperbolic polyhedron, and let $col$ be any $r$-admissible coloring of the edges of $\Gamma$. The sharp upper-bound conjecture.

$$\frac{\pi}{r}\log\left\lvert Y_r(\Gamma,col)\right\rvert\leq \operatorname{Vol}(\overline{\Gamma})+O\left(\frac{\log(r)}{r}\right).$$

Moreover, the inequality is sharp, with equality attained by the sequence of colorings assigning $\frac{r-2\pm1}{2}$ to every edge, with the sign chosen so that the colors are even. This is presented as a sharper conjecture following the maximum volume conjecture. Its general status is not resolved in the supplied text.