The pseudo-Abelian variety conjecture for coframed projective manifolds

Let $X$ be a $k$-coframed projective manifold, let $q=h^0(\Omega^1_X)$, and suppose that $q=k$. Assume also that the canonical divisor $K_X$ is nef. Pseudo-Abelian variety conjecture. Then $X$ is a pseudo-Abelian variety. This extends the preceding theorem beyond its currently established cases; the assertion would follow in particular from the Abundance conjecture, and remains open in general.