Ellipsoidal positivity-set conjecture for cubic thin-obstacle solutions

Let $u$ be a global solution to the thin obstacle problem with cubic growth, and suppose that its positivity set on the thin space is bounded. Cubic positivity-set conjecture. The positivity set on the thin space is convex, or more specifically, it is an ellipsoid. The known result in this setting gives only that the positivity set is star-shaped at some point; whether it must be convex or ellipsoidal remains open.