Projective categoricity conjecture for compact convex bodies

Let $K,K'$ be compact convex bodies in $\mathbb{R}^n$, with $n\ge2$. Write $K\equiv_{\mathcal{L}_{\mathrm{B}}}K'$ when they satisfy the same sentences in the projective betweenness language, and call them projectively equivalent when a projective transformation maps one onto the other.

Projective categoricity conjecture. $K\equiv_{\mathcal{L}_{\mathrm{B}}}K'$ if and only if $K$ and $K'$ are projectively equivalent.

This is the projective analogue of the proved affine categoricity theorem. The paper explains that the affine reconstruction transposes only partly to the projective setting; the conjecture remains open.