No quantum symmetry for O^-(6, q) with q > 3

Prove that the orthogonal polar graphs O^-(6, q) for all prime-power values q > 3 do not have quantum symmetry; equivalently, establish that the quantum automorphism group Aut^+(O^-(6, q)) is commutative and coincides with the classical automorphism group of O^-(6, q).

Background

The paper determines quantum automorphism groups for all 3-transitive graphs except the orthogonal polar graphs O^-(6, q) with q > 3. It proves that there is no quantum symmetry for the McLaughlin graph and for O^-(6, q) with q = 2 or 3, and it identifies monoidal equivalences for the affine polar graphs VO^{+}(2k,2) and VO^{-}(2k,2). The remaining unresolved case concerns O^-(6, q) for q > 3.

Motivated by partial progress toward a general proof, the authors formulate a conjecture that O^-(6, q) for q > 3 also has no quantum symmetry. They note that several steps of their proof for q = 2 extend to general q, but they were unable to complete the argument in full generality.