Algorithmic conjecture for isotypic multiplicities of symmetric semi-algebraic sets

Fix $d>0$. Let $S\subset \mathbb{R}^k$ be a symmetric semi-algebraic set defined by a $\mathcal{P}$-closed formula, where $\mathcal{P}$ is a set of symmetric polynomials of degrees bounded by $d$. For each partition $\lambda\vdash k$, write $m_{i,\lambda}(S,\mathbb{Q})$ for the multiplicity of the irreducible representation indexed by $\lambda$ in the $i$th cohomology module, and write $b_i(S,\mathbb{Q})$ for the corresponding Betti numbers.

Algorithmic conjecture. There is an algorithm which computes $m_{i,\lambda}(S,\mathbb{Q})$ for every $\lambda\vdash k$ with $m_{i,\lambda}(S,\mathbb{Q})>0$, as well as all the Betti numbers $b_i(S,\mathbb{Q})$, with complexity polynomial in $\operatorname{card}(\mathcal{P})$ and $k$.

The preceding polynomial bounds on the number of representations and on their multiplicities motivate this conjecture, while the hook-length formula makes the dimensions of the relevant Specht modules polynomially computable.