Hu's rational cycle class conjecture for Chow varieties

Let $C_{p,d}({\mathbb{P}}^n)$ denote the Chow variety of effective $p$-dimensional cycles of degree $d$ in projective space, and let ${\rm Ch}_q(-)_{\mathbb{Q}}$ denote the Chow group of $q$-dimensional cycles with rational coefficients. Let $H_{2q}(-,{\mathbb{Q}})$ denote singular homology with rational coefficients.

Hu's rational cycle class conjecture. For all $d\geq 0$ and $0\leq p\leq n$,

$${\rm Ch}_q(C_{p,d}({\mathbb{P}}^n))_{\mathbb{Q}}\cong H_{2q}(C_{p,d}({\mathbb{P}}^n),{\mathbb{Q}}),\qquad \forall q\geq 0.$$

This is proposed as a weaker version of the integral conjecture; the source states that the corresponding cycle class assertion is known after tensoring with rational coefficients, while the full conjecture remains open.