Mumford's conjecture with equivalent uniruledness and curvature conditions

Let $X$ be a projective manifold. Mumford's equivalent-conditions conjecture. If

$$H^0(X,(T_X^*)^{\otimes m})=0$$

for every $m\geq 1$, then each of the following holds: $X$ is uniruled; $K_X$ is not pseudo-effective; $K_X^{-1}$ is RC-positive; $X$ admits a Hermitian metric with positive Chern scalar curvature; and, for every ample line bundle $A$, there is a positive integer $c_A$ such that

$$H^0(X,K_X^{\otimes \ell}\otimes A^{\otimes k})=0$$

for $\ell\geq c_A(k+1)$ and $k\geq 0$. The source explains that the uniruledness conjecture would imply this formulation, so it remains open insofar as that conjecture remains open.