  Using recent work of Bettiol, we show that a first-order conformal
deformation of Wilking's metric of almost-positive sectional curvature on
$S^2\times S^3$ yields a family of metrics with strictly positive average of
sectional curvatures of any pair of 2-planes that are separated by a minimal
distance in the 2-Grassmanian. A result of Smale's allows us to conclude that
every closed simply connected 5-manifold with torsion-free homology and trivial
second Stiefel-Whitney class admits a Riemannian metric with a strictly
positive average of sectional curvatures of any pair of orthogonal 2-planes.
