Diagonal Ricci flow conjecture for cohomogeneity one manifolds

Let $(M,\mathsf{G})$ be a cohomogeneity one manifold, let $\mathcal{B}$ be a stably Ricci-diagonal basis for it, and let $\mathrm{g}_0$ be a metric on $M$ that is diagonal with respect to $\mathcal{B}$. Diagonal Ricci flow conjecture. The Ricci flow evolving metric $\mathrm{g}(t)$ is also diagonal in the basis $\mathcal{B}$. If true, this would show that diagonal invariant metrics remain diagonal under Ricci flow, yielding a time-independent frame that diagonalizes the metric restriction on each orbit and preserving the corresponding geodesic property for transverse curves. The claim is motivated by the necessity of stable Ricci-diagonality for avoiding off-diagonal terms to first order, but preservation at all later times is not established here.