Does smashing of a pointed CW complex $X$ with an arbitrary pointed CW complex $Y$ increase the connectivity?
The connectivity of a pointed space $X$ is the maximal number $\operatorname{con}(X)$ such that $\pi_i(X)=0$ for all $0\leq i\leq\operatorname{con}(X)$.
More precisely, the question is: $\operatorname{con}(X\wedge Y)\geq\operatorname{con}(X)$?