Dimension-decomposition 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 dimension decomposition $(n_1,n_2)$, where $n_1\geq n_2>1$. Dimension-decomposition conjecture. If $F$ is restrictively CW-homogeneous, then it must be Riemannian. This is presented as a stronger formulation of the preceding conjecture, replacing the structural condition on the commutative subalgebra with the dimension condition. Its resolution is not specified in the supplied text.