Polynomial growth conjecture for Specht multiplicities of symmetric real algebraic sets

Let $\Lambda_n=\mathrm{R}[X_1,\ldots,X_n]^{\mathfrak{S}_n}$ be the graded ring of invariant polynomials, let $\Lambda=\varprojlim \Lambda_n$, and let $I=(f_1,\ldots,f_k)$ be a finitely generated ideal of $\Lambda$. Write

$$V_n(I)=\operatorname{Zer}(\phi_n(f_1),\ldots,\phi_n(f_k))\subset \mathrm{R}^n$$

for the associated symmetric real algebraic sets. For a fixed partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)\vdash d$, set $\{\lambda\}_n=(n-d,\lambda_1,\ldots,\lambda_\ell)$ for $n\geq \lambda_1+d$, and let $m_{p,\{\lambda\}_n}(V_n(I))$ denote the multiplicity of the Specht module indexed by $\{\lambda\}_n$ in the relevant cohomology module of $V_n(I)$.

Polynomial growth conjecture. For any fixed $p\geq 0$, $m_{p,\{\lambda\}_n}(V_n(I))$ is eventually given by a polynomial in $n$.

This conjecture predicts polynomial growth for the multiplicities of stable Specht modules in the cohomology of sequences of symmetric real algebraic sets, extending the representational stability phenomenon to this setting. Its resolution status is not established by the supplied source material.