Lawson homology conjecture for symmetric products of projective spaces

Let $\mathrm{SP}^d(\mathbb{P}^n)$ be the $d$-th symmetric product of complex projective space, and let $cl$ denote the cycle class map from Lawson homology to singular homology with integer coefficients. Symmetric-product Lawson homology conjecture. For all $d\geq 0$, $q\geq 0$, and $0\leq p\leq n$, the map

$$cl:L_qH_k(\mathrm{SP}^d(\mathbb{P}^n))\stackrel{\cong}{\longrightarrow}H_k(\mathrm{SP}^d(\mathbb{P}^n),\mathbb{Z})$$

is an isomorphism. This is proposed as the $p=0$ special case of the preceding conjecture; the source notes that it is more likely for $0\leq q\leq d$ and that the corresponding statement holds after tensoring with rational coefficients.