Mixed-Hodge-polynomial conjecture for modular data

Let $\mathcal{M}_{g,n}^{SU(2)/\mathbb{Z}_2}$ be the relevant moduli space, and let

$$P_{g,n}(q)=\sum_i a_iq^{d_i}$$

be the renormalization of the pure part of its mixed Hodge polynomial. Let $V_{g,n}$ be the associated vertex operator algebra, let $\mathbb{V}^{\mathrm{mod}}_{g,n}$ be its modular representation, and let $[d_i^{a_i}]$ denote Jordan blocks of size $d_i$ with multiplicity $a_i$.

Mixed-Hodge-polynomial conjecture. The total number of fixed manifolds of $\mathcal{M}_{g,n}$, equivalently the number of simple modules of $V_{g,n}$, is

$$P_{g,n}(1)=\sum_i a_i,$$

the modular representation has dimension

$$\dim\mathbb{V}^{\mathrm{mod}}_{g,n}=\sum_i a_id_i,$$

and the Jordan type of the modular matrix is $[d_i^{a_i}]$.

This conjecture proposes that the renormalized pure mixed Hodge polynomial simultaneously encodes the number of simple modules, the modular-representation dimension, and its Jordan type. The source does not state a resolution, so it remains open.