Separable essential finite-generation conjecture for valuation rings

Let $(K,\nu)\subset (L,\omega)$ be in situation, with valuation rings $\mathcal{O}_\nu$ and $\mathcal{O}_\omega$. Assume in addition that the field extension $L|K$ is separable. Separable essential finite-generation conjecture. The valuation ring $\mathcal{O}_\omega$ is essentially finitely generated over $\mathcal{O}_\nu$. The paper states that proving this separable case is enough to prove the preceding conjecture; no resolution is given in the supplied text.