A hereditarily Baire space is a topological space having the property that each of its closed non-empty subspaces is of second category. J. M. Aarts and D. J. Lutzer [Proc. Amer. Math. Soc. 38 (1973), 198--200; MR0309056 (46 #8167)] raised the question of whether the Cartesian product of a compact space $X$ with a hereditarily Baire space $Y$ must remain hereditarily reflexive. The present paper establishes this conjecture when $X$ is Čech complete and $Y$ is Hausdorff, regular, and the image of a metrizable space under a closed-continuous map. The last condition can be replaced by assuming that each closed non-empty subset of $Y$ has a point with a countable (relative) neighborhood base