Productive Menger and Hurewicz properties imply their weak counterparts

Let $X$ be a topological space. Say that $X$ is productively Menger if $X\times Y$ is Menger for every Menger space $Y$, and productively Hurewicz if $X\times Y$ is Hurewicz for every Hurewicz space $Y$. Likewise, define productively weakly Menger and productively weakly Hurewicz by requiring the corresponding product to be weakly Menger or weakly Hurewicz for every space with that property.

Productive weak-property conjectures. If a space is productively Menger then it is productively weakly Menger. If a space is productively Hurewicz then it is productively weakly Hurewicz.

These questions concern whether productive versions of the classical covering properties imply productive versions of their weaker analogues. The source presents them as conjectures whose resolution may depend on characterizations of the relevant productive properties.