Positive-relatedness conjecture for Riemannian g.o. metrics

Let $g_1,\dots,g_k$ be a family of positively related Riemannian metrics on a homogeneous space. A family is positively related when its metrics admit a common decomposition

$$g_j=\sum_{i=1}^s a_{ji}\alpha_i,$$

with $a_{ji}>0$ for all $i,j$ and fixed invariant scalar products $\alpha_i$ on the irreducible summands. Positive-relatedness conjecture. If one metric of this family is a geodesic-orbit (g.o.) metric, then every metric in the family is also a g.o. metric. This proposes that the g.o. property is shared throughout a positively related family, although the source notes that a proof in full generality does not seem easy.