Let $H$ be a separable Hilbert space. Let $A \subseteq B(H)$ be a finitely generated unital algebra. Let $M$ be the strong operator topology closure of $A.$ Let $B_A$ be the closed ball in $A$ and $B_M$ be the closed ball in $M.$ Is $B_M$ the strong operator topology closure of $B_A?$ (More likely, is there a counterexample?)