I'm trying to prove the following:

Let $L$ be a positive holomorphic line bundle on a compact complex manifold $X$. For any hermitian holomorphic vector bundle $E$ on $X$, there is $k \in \mathbb{N}$ such that $E\otimes L^k$ is Nakano positive.

I'm using the decomposition given by diverietti in this answer. Part of my problem is that (I think) I can find a sufficiently large choice of $k$ so that $E\otimes L^k$ is Nakano positive at a point, but I can't extend beyond that. If I try to prove that I can find such a $k$ locally, my bounds are no longer valid. Maybe I need to use the fact that $X$ is locally compact.

Does anyone have a reference (or proof) of the above fact?