Commutation criterion for compatible vector bundle structures on homogeneity manifolds

Let $(E,\nabla)$ be a homogeneity manifold, and let a vector bundle structure on $E$ have Euler vector field $\nabla_E$. Compatibility means that this vector bundle structure is compatible with the homogeneity structure on $(E,\nabla)$. Commutation conjecture. The vector bundle structure is compatible if and only if

$$[\nabla,\nabla_E]=0.$$

This would characterize compatibility of homogeneity vector superbundles by commutation of their weight vector fields, paralleling the characterization of double vector bundles. It is stated as an open question.