Local convexity obstructions for bounded realizations of Teichmüller space

Let $D\subset\mathbb{C}^n$, $n\ge2$, be a bounded domain that is biholomorphic to a finite-dimensional Teichmüller space. We prove that $\partial D$ has no locally convex boundary patch containing a complex $(n-1)$-dimensional manifold through each of its points. Note that in complex dimension two this excludes every locally convex boundary point, without any boundary regularity assumption. In every dimension, we also exclude locally convex $C^2$ boundary patches.