Real one-orbit Gram graph conjecture

Let $V$ be a finite-dimensional real Hilbert space, and let $G\leq \operatorname{O}(V)$ be finite. For a generic $x\in V$, consider the Gram graph of the orbit $Gx$. Real one-orbit Gram graph conjecture. The isomorphism class of the Gram graph of $Gx$ determines $G$ up to isomorphism. This is the proposed real analogue of the one-orbit theorem for finite subgroups of the unitary group; the source states that the claim is believed to hold, but that a proof is not known.