The exotic sphere pinching conjecture

An exotic sphere is a smooth manifold homeomorphic but not diffeomorphic to a standard sphere. A Riemannian metric is strictly $1/4$-pinched in the global sense when its sectional curvature satisfies $\frac14<K\leq 1$. Exotic sphere pinching conjecture. An exotic sphere cannot admit a metric with $1/4$-pinched sectional curvature. The conjecture concerns whether the Topological Sphere Theorem can be strengthened from homeomorphism to diffeomorphism; the source gives no resolution.