Riemannianity conjecture for restrictively CW-homogeneous left invariant $(\alpha_1,\alpha_2)$-metrics

Let $F$ be a left invariant $(\alpha_1,\alpha_2)$-metric on a compact connected simple Lie group $G$, with a decomposition $\mathfrak{g}=\mathbf{V}_1+\mathbf{V}_2$ such that $\mathbf{V}_2$ is a commutative subalgebra. Riemannianity conjecture. If $F$ is restrictively CW-homogeneous, then it must be Riemannian. This conjecture generalizes the two preceding theorems, which establish the conclusion when $\mathbf{V}_2$ is a Cartan subalgebra of dimension greater than one or a two-dimensional commutative subalgebra. The general case remains unresolved in the supplied text.