On the stability of free boundary minimal submanifolds in conformal domains

Given a $n$-dimensional Riemannian manifold with non-negative sectional curvatures and convex boundary, that is conformal to an Euclidean convex bounded domain, we show that it does not contain any compact stable free boundary minimal submanifold of dimension $2\le k\le n-2$, provided that either the boundary is strictly convex with respect to any of the two metrics or the sectional curvatures are strictly positive.