Alev's Poisson–Hochschild homology dimension conjecture for types B and D

Let $S^W$ be the invariant polynomial algebra associated with the Weyl group $W$, and let $HP_0(S^W)$ and $HH_0(S^W)$ denote its zeroth Poisson and Hochschild homology spaces. For type $B_n$, write $bpi(n)$ for the number of partitions of $n$, and for type $D_n$, write $\widetilde{\pi}(n)$ for the number of partitions of $n$ having an even number of parts.

Alev's conjecture.

$$\dim HP_0(S^W)=\pi(n)\quad\text{for type }B_n,$$

and

$$\dim HP_0(S^W)=\widetilde{\pi}(n)\quad\text{for type }D_n.$$

The corresponding Hochschild-homology dimensions are known from the Alev–Farinati–Lambre–Solotar theorem, and the conjecture asserts that the Poisson-homology dimensions agree with them. The paper verifies the claim in the cases $B_2$, $D_2$, $B_3$, and $D_3$.