The inverse-linear conjecture for nonnegatively curved left-invariant metrics

Let $G$ be a compact Lie group, let $h_0$ be a fixed bi-invariant metric on $G$, and let $h=h_1$ be a left-invariant metric with nonnegative curvature. The inverse-linear path from $h_0$ to $h_1$ is the unique inverse-linear path joining these metrics.

Inverse-linear conjecture. If $h$ has nonnegative curvature, then the inverse-linear path from $h_0$ to $h_1=h$ consists entirely of metrics with nonnegative curvature.

Inverse-linear paths arise naturally from families of submersion metrics, and the conjecture proposes a broad mechanism for producing nonnegatively curved metrics. The supplied text gives supporting evidence but does not state that the claim has been proved or disproved.