The Chow group and Lawson homology conjecture for Chow varieties of projective space

Let $C_{p,d}({\mathbb{P}}^n)$ denote the Chow variety of effective $p$-dimensional algebraic cycles of degree $d$ in projective space. Write $\operatorname{Ch}_q$ for the codimension-$q$ Chow group, $L_qH_k$ for Lawson homology, and $H_k(-,\mathbb{Z})$ for singular homology with integer coefficients. The Chow group and Lawson homology conjecture. For $d\geq 0$ and $0\leq p\leq n$,

$$\operatorname{Ch}_q(C_{p,d}({\mathbb{P}}^n))\cong H_{2q}(C_{p,d}({\mathbb{P}}^n),\mathbb{Z})$$

for all $q\geq 0$, and

$$L_qH_k(C_{p,d}({\mathbb{P}}^n))\cong H_k(C_{p,d}({\mathbb{P}}^n),\mathbb{Z})$$

for all $k\geq 2q\geq 0$. The conjecture would identify the Chow groups and Lawson homology of these generally singular parameter spaces with their singular homology, extending the proposition established in the paper for $0$-cycles and for $L_1H_k$; the general statement remains open.