Rational Lawson homology conjecture for Chow varieties

Let $C_{p,d}(\mathbb{P}^n)$ denote the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective space, let $L_qH_k(-)_{\mathbb{Q}}$ denote Lawson homology with rational coefficients, and let $H_*(-,\mathbb{Q})$ denote singular homology with rational coefficients. Rational Lawson homology conjecture. For all $d\geq 0$ and $0\leq p\leq n$,

$$L_qH_k(C_{p,d}(\mathbb{P}^n))_{\mathbb{Q}}\cong H_k(C_{p,d}(\mathbb{P}^n),\mathbb{Q}),\qquad \forall k\geq 2q\geq 0.$$

This is presented as a weaker version of the integral conjecture, and the source explains that the assertion means that all even-dimensional topological cycles on Chow varieties are algebraic with rational coefficients.