Eisenhart-type product conjecture for affinely nonrigid sub-Riemannian metrics

Let $(M,D,g)$ be a sub-Riemannian structure, and call $g$ affinely rigid if the sub-Riemannian metrics constantly proportional to it are the only metrics on $(M,D)$ affinely equivalent to $g$. A sub-Riemannian structure admits a product structure if, locally, it is a product of two sub-Riemannian structures on manifolds of positive dimension and with distributions of positive rank.

Eisenhart-type conjecture. If a sub-Riemannian metric $g$ is not affinely rigid near a point $q_0$, meaning that it admits a locally affinely equivalent non-constantly proportional sub-Riemannian metric in a neighborhood of $q_0$, then $g$ is the direct product of two sub-Riemannian metrics in a neighborhood of $q_0$.

This is the proposed local analogue of Eisenhart's theorem for Riemannian metrics. The preceding product construction shows that sub-Riemannian metrics admitting a product structure are affinely nonrigid; the conjecture asks whether all locally affinely nonrigid metrics arise in this way.