Rationality of isotypic Hilbert series for nonzero symmetric ideals

Let $K[X]$ be the polynomial ring in countably many variables, with the symmetric group acting by permuting variables. A homogeneous ideal $I\subset K[X]$ is symmetric if it is invariant under this action, and $H^\mu(K[X]/I;t)$ denotes the isotypic Hilbert series associated with a partition $\mu$. Rationality conjecture. If $I\neq (0)$, then, for every partition $\mu$, the isotypic Hilbert series

$$H^\mu(K[X]/I;t)$$

is a rational function. The zero ideal is an exceptional counterexample: its symmetric and isotypic Hilbert series are not rational, whereas the conjecture predicts rationality for every nonzero homogeneous symmetric ideal. The claim is motivated by the absence of further counterexamples and by rationality results obtained for several classes of symmetric ideals, including ideals containing a hook Specht ideal; a general structural explanation remains open.