King's tilting bundle conjecture for smooth complete geometric quotients

Let $V$ be a vector space, let $G$ be a reductive group acting linearly on $V$, and let $U\subseteq V$ be a Zariski-open subset such that the geometric quotient $U/G$ is a smooth complete variety. King's tilting bundle conjecture. The variety $U/G$ has a tilting bundle. This conjecture is motivated by quiver flag varieties, which provide examples of smooth geometric quotients with explicit tilting bundles even when they are not Fano; its general status is not specified here.