This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in absolute value by 1. Kotschick proved, in Entropies, volumes, and Einstein metrics (published version), that there are distinct smooth structures on $k(S^2 \times S^2) \sharp (1 + k)(S^1 \times S^3)$, $k$ sufficiently large, for which in the standard smooth structure, the minimal volume $=0$ (by finding a fixed-point free circle action), and another smooth structure for which the minimal volume is bounded away from 0.
My question is whether the converse is true: if there is a metric in which the minimal volume $=0$, must the smooth structure be standard? The existence of a polarized F-structure in this case may be relevant.