The Frobenius–Perron dimension conjecture for distinguished nilpotent categories

Let $\mathfrak{g}$ be a Lie algebra, let $e\in\mathfrak{g}$ be distinguished nilpotent, and let $Q=Q(e)$ be the finite centralizer of an $sl_2$-triple associated with $e$ in the simply connected group $G$ with $\operatorname{Lie}(G)=\mathfrak{g}$. If the Jordan cell sizes for the adjoint action of $e$ on $\mathfrak{g}$ are $2k_i+1$ for $i\in M=M(e)$, define

$$S_k(l):=\frac{l}{\left(2^k\sin(\pi/l)\sin(2\pi/l)\cdots\sin(k\pi/l)\right)^2}.$$

The distinguished-nilpotent Frobenius–Perron dimension conjecture.** If $m=1$ or $l$ is undivisible, then

$$\operatorname{FPdim}(\mathcal{C}(\mathfrak{g},e,l,q))=|Q|\prod_{i=1}^{M}S_{k_i}(l).$$

This specializes to the stated $G_2$ and $C_3$ examples. The source gives supporting computations and a heuristic explanation, but no general proof; the divisible case for $m>1$ is explicitly left without a formula.