Invariant-distribution conjecture for volume-preserving vector fields

Let $M$ be a closed, connected, orientable manifold, and let $X$ be a volume-preserving vector field on $M$. Write $\mathcal I_X(M)$ for the space of all $X$-invariant distributions. Invariant-distribution conjecture. If

$$\dim \mathcal I_X(M)=1,$$

then $M$ is diffeomorphic to a torus. This conjecture is motivated by the observation that known volume-preserving uniquely ergodic vector fields that are not cohomology free generally have many invariant distributions, with constant Liouville vector fields on tori as an exception. The conjecture is presented as a proposed direction and remains open.