Dolbeault Cohomology Growth under Curvature Rank Bounds

Let $L$ be a smooth Hermitian holomorphic line bundle on a compact complex manifold $X$, and let $r$ be the maximal rank of its curvature. We prove that $h^q(X,L^p\otimes E)=O_\varepsilon(p^{r+\varepsilon})$ for every fixed holomorphic vector bundle $E$, every $\varepsilon>0$, and every degree $q$.