The product decomposition conjecture for 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-zero section with $q<\dim X$. A finite étale cover is a finite surjective morphism that is étale. The product decomposition conjecture. There exists a finite étale cover $\tilde X \to X$ such that $\tilde X \simeq Y \times Z$, where $Y$ is a projective manifold with trivial canonical bundle and $v$ is induced by a section $v' \in H^0(Y,\bigwedge^q T_Y \otimes L')$. This generalizes the known decomposition phenomenon for nowhere-vanishing vector fields, but the stated higher-rank version is open.