Orthogonal free quantum group factors are strongly 1-bounded

We prove that the orthogonal free quantum group factors L(FON) are strongly 1-bounded in the sense of Jung. In particular, they are not isomorphic to free group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the quantum Cayley tree associated to FON, and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.