Product selection-principle theorem for spaces of countable selection rank

For a metric space $X$, let ${\sf tp}_{{\sf S}_1(\mathcal{O},\mathcal{O})}(X)$ denote its associated selection-principle type, and suppose this type is countable. Let $Y$ be a topological space. Here ${\sf S}_c(\mathcal{O},\mathcal{O})$, ${\sf S}_{fin}(\Omega,\mathcal{O}^{gp})$, and ${\sf S}_1(\Omega,\mathcal{O}^{gp})$ are the selection properties appearing below.

Product selection-principle theorem. If $Y$ has ${\sf S}_c(\mathcal{O},\mathcal{O})$ and ${\sf S}_{fin}(\Omega,\mathcal{O}^{gp})$, then $X\times Y$ has ${\sf S}_c(\mathcal{O},\mathcal{O})$. If $Y$ has ${\sf S}_{fin}(\Omega,\mathcal{O}^{gp})$, then $X\times Y$ has ${\sf S}_{fin}(\mathcal{O},\mathcal{O})$. If $Y$ has ${\sf S}_1(\Omega,\mathcal{O}^{gp})$, then $X\times Y$ has ${\sf S}_1(\mathcal{O},\mathcal{O})$.

The result gives product-preservation statements for several covering and selection properties. The supplied text presents it as a conjecture-environment candidate, but does not establish its resolution status; the surrounding proof discussion suggests that the assertions may in fact be proved in the paper.