Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly generated.
Let $E$ be a Banach space with the norm $\|\cdot\|$ and the unit ball $B_E$. Let $|||\cdot|||\le \|\cdot\|$ be another norm, and let $\tau$ be a linear (or even locally convex) topology which is stronger than the $|||\cdot|||$-topology, but weaker than the $\|\cdot\|$-topology on $B_E$. Does it follows that $B_E$ with $\tau$ is compactly generated?