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 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.