Characterization of the $CD(0,m)$-condition by monotonicity of $W_m$-entropy

Let $(M,g)$ be a compact Riemannian manifold, or a complete Riemannian manifold with bounded geometry, and let f\begin{in}C^\infty(M)\end{in} satisfy $\nabla f\in C_b^\infty(M)$. Suppose that the $W_m$-entropy associated with either the heat equation of the Witten Laplacian or the optimal transport problem is non-decreasing for $t\in[0,T]$.

$CD(0,m)$ characterization conjecture. Then the $CD(0,m)$-condition holds, that is,

$$\operatorname{Ric}_{m,n}(L)\geq 0.$$

This conjecture proposes a converse to the known implication from the $CD(0,m)$-condition to convexity and entropy monotonicity, characterizing the curvature-dimension condition on complete Riemannian manifolds through monotonicity of the $W_m$-entropy. Its status is not resolved in the supplied source.