Productively Scheepers spaces and their relatives

We prove that assuming $\mathfrak{b}=\mathfrak{d}$, in the class of hereditarily Lindel\"of spaces, each productively Scheepers space is productively Hurewicz. The above statement remains true in the class of all general topological spaces assuming that $\mathfrak{d}=\aleph_1$. To this end we use combinatorial methods and the Menger covering property parametrized by ultrafilters. We also show that if near coherence of filters holds, then the Scheepers property is equivalent to a Menger property parametrized by any ultrafilter.