Almost positive curvature on the Gromoll-Meyer sphere

Gromoll and Meyer have represented a certain exotic 7-sphere $\Sigma^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $\Sigma^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\geq 1$ which we describe explicitly.