Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $F\to X$ be a holomorphic vector bundle, and let $L\to X$ be a positive holomorphic line bundle. Then there exists an integer $m_0\ge 1$ such that, for every $m\ge m_0$, the holomorphic vector bundle $F\otimes L^m$ admits a Hermitian metric whose Chern curvature is Nakano positive. Consequently, for every $q\ge 1$ and every $m\ge m_0$,

\begin{align*} H^q(X,K_X\otimes F\otimes L^m)=0. \end{align*}