Hu's integral 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 ${\rm Ch}_q(-)$ denote the Chow group of $q$-dimensional cycles. For $d\geq 0$, $0\leq p\leq n$, and $q\geq 0$, let $H_{2q}(-,{\mathbb{Z}})$ denote singular homology with integer coefficients.

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

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

The conjecture would identify algebraic cycles on every Chow variety with the corresponding even-dimensional integral homology classes. The source notes that the integral statement may be too optimistic, although no evidence is known that the cycle class map fails to be injective or surjective.