Weak precompactness in projective tensor products

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact in the projective tensor product $X \widehat{\otimes}_\pi Y$, then either $W_1$ or $W_2$ is relatively norm compact. For instance, such a property holds for the pair $(\ell_p,\ell_q)$ if $1<p,q<\infty$ satisfy $1/p+1/q\geq 1$. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of $\ell_1$ into $X \widehat{\otimes}_\pi Y$ for arbitrary Banach spaces $X$ and $Y$, in connection with the compactness of all operators from $X$ to $Y^*$.