Hörmander nondegeneracy conjecture for irreversible drift

Consider the stochastic differential equation

$$dx=V_0(x)\,dt+\sum_{i=1}^{m}V_i(x)\circ dW_i$$

with smooth vector fields on $\mathbf{R}^n$, and let $\mathscr{V}_k$ be the recursively defined families of vector fields used in the parabolic Hörmander condition. Write $\operatorname{Im}\sigma(x)$ for the image of the volatility matrix at $x$, and let $b_{\mathrm{irrev}}$ denote the irreversible component of the drift. The equation is hypoelliptic when the parabolic Hörmander condition holds everywhere.

Hörmander nondegeneracy conjecture. If the parabolic Hörmander condition holds everywhere, then it cannot be the case that $b(x)\in\operatorname{Im}\sigma(x)$ for any $x$; equivalently, the irreversible drift should not belong to the image of the volatility on any open set.

The proposed implication is motivated by the support theorem and by the observation that if the drift lies in the image of the volatility on an open set, the process should fail to span an open set there. The source presents this as an expectation rather than a proved result, and the notation and exact relation between $b$ and $b_{\mathrm{irrev}}$ should be checked.