Projective categoricity conjecture for the ball

Let $K\subseteq\mathbb{R}^n$ be a compact convex body, with $n\ge2$. Let $\overline{B}{}^{\,n}$ denote the closed unit ball, and write $K\equiv_{\mathcal{L}_{\mathrm{B}}}\overline{B}{}^{\,n}$ for elementary equivalence in the projective betweenness language. A quadric is a set whose boundary is a quadric hypersurface; projective equivalence means that a projective transformation maps one body onto the other.

Ball categoricity conjecture. $K\equiv_{\mathcal{L}_{\mathrm{B}}}\overline{B}{}^{\,n}$ if and only if $\partial K$ is a quadric, equivalently, if and only if $K$ is projectively equivalent to $\overline{B}{}^{\,n}$.

The conjecture gives a proposed geometric characterization of the elementary theory of the ball. The paper proves that the ball and certain quartic bodies are not elementarily equivalent, but the full characterization remains open.