The virtual Hodge and Betti number conjecture for Chow varieties of projective space

For integers $n\geq p\geq 0$ and $d\geq 0$, let $C_{p,d}({\mathbb{P}}^n)$ denote the Chow variety of effective $p$-dimensional algebraic cycles of degree $d$ in projective space, let $\tilde{h}^{r,s}$ denote its virtual Hodge numbers, and let $\tilde{\beta}^{i}$ denote its virtual Betti numbers. The virtual Hodge and Betti number conjecture.

$$\tilde{h}^{r,s}(C_{p,d}({\mathbb{P}}^n))=0$$

for all $r\neq s$. In particular,

$$\tilde{\beta}^{i}(C_{p,d}({\mathbb{P}}^n))=0$$

for $i$ odd. This extends the known vanishing of the off-diagonal virtual Hodge sums and the first virtual Betti-number identities, but the structure of these Chow varieties is not sufficiently understood to establish the full assertion.