Projective categoricity conjecture for open properly convex sets

Let $U,V\subseteq\mathbb{R}^n$ be open, properly convex subsets. Write $U\equiv_{\mathcal{L}_{\mathrm{B}}}V$ when they are elementarily equivalent in the projective betweenness language, and call them projectively equivalent when a projective transformation maps one onto the other.

Open-domain projective categoricity conjecture. Two such sets $U$ and $V$ are $\mathcal{L}_{\mathrm{B}}$-elementarily equivalent if and only if they are projectively equivalent.

This extends the compact-body projective categoricity conjecture to properly convex open domains. Its resolution would connect elementary equivalence in the betweenness language with the projective classification of Hilbert geometries; the extension remains open.