The conjecture on zeros of twisted multivector fields

Let $X$ be a projective manifold, let $L$ be a numerically trivial line bundle, and let $v \in H^0(X,\bigwedge^q T_X \otimes L)$ be a non-trivial section. A zero of $v$ is a point where its value vanishes. The zero-free multivector-field conjecture. If $v$ has a zero, then $X$ is uniruled. The assertion is clear when $q=\dim X$, and the case $q=1$ with $L$ trivial is classical; the general statement remains open in the source.