Tapp's conjecture on inverse-linear paths of nonnegatively curved metrics

Let $G$ be a compact Lie group with a bi-invariant metric $h_0$ and Lie algebra $\mathfrak g$. For a left-invariant metric $h$ on $G$, let $\Phi$ be the unique $h_0$-self-adjoint positive definite endomorphism satisfying $h(X,Y)=h_0(\Phi X,Y)$, and let $\Phi_t=(I-t\Psi)^{-1}$ be the unique inverse-linear path with $\Phi_1=\Phi$. Such a path is called nonnegative when the corresponding metrics $h_t$ are nonnegatively curved for $0\leq t\leq 1$. Tapp's conjecture. If $h$ is a nonnegatively curved left-invariant metric on $G$, then the unique inverse-linear path from $h_0$ to $h$ is nonnegative. This conjecture would give a systematic way to study nonnegatively curved left-invariant metrics through inverse-linear variations, but its resolution is not established in the supplied source.