Value-variety decomposition conjecture for multisymmetric isotypic Hilbert series

Let $K[X^1,\dots,X^k]$ be the polynomial ring in $k$ sets of countably many variables, with $\operatorname{Sym}(\infty)$ permuting the indices. Let $I\subset K[X^1,\dots,X^k]$ be a homogeneous $\operatorname{Sym}(\infty)$-invariant ideal. For a point $p=(p_i)_{i\ge1}\in V(I)(\overline K)$, let $V_p\subseteq\mathbb{A}^k_{\overline K}$ be the Zariski closure of the set of values $\{p_i\mid i\ge1\}$, called its value variety, and set

$$\dim V(I):=\sup_p\dim V_p.$$

For $d\ge0$, define

$$P_d(t):=\prod_{m\ge1}(1-t^m)^{-\binom{m+d-1}{d-1}},$$

with $P_0=1$. Value-variety decomposition conjecture. For every partition $\mu$,

$$H^{\mu}\bigl(K[X^1,\dots,X^k]/I;t\bigr)=\sum_j R_j(t)\prod_i P_{d_{ji}}(t),$$

where $R_j(t)\in\mathbb{Q}(t)$ and each $d_{ji}\ge1$ is the dimension of a positive-dimensional component of a value variety of $V(I)$. In particular, $H^\mu$ is rational for every $\mu$ if and only if $\dim V(I)=0$. This conjectural description would relate the transcendental factors in the Hilbert series to the dimensions of value varieties; the preceding examples support this picture, but the general decomposition and its rationality criterion remain unproved.