Conjecture on quasi Frobenius groups in o-minimal structures

Let $C<G$ be a connected quasi Frobenius group definable in an o-minimal structure, with $C$ definable and connected. Assume that $G$ contains involutions, and that

$$\bigcup_G C^g$$

contains all translations, that is, products of two involutions. The o-minimal quasi Frobenius conjecture. Then $G$ is isomorphic to

$$\operatorname{SO}_3(R)$$

for a real closed field $R$. The conjecture seeks to characterize the classical group $\operatorname{SO}_3(R)$ from the geometry of involutions in the o-minimal setting; the supplied text gives no evidence that it has been resolved.