If there would be such a quantum effect it would contradict a lot of conventional theoretical physics, brains operate at high temperatures ~310 Kelvin, and the

If there would be such a quantum effect it would contradict a lot of conventional theoretical physics, brains operate at high temperatures ~310 Kelvin, and the length scales involved are fairly large for quantum mechanics. You would only expect to see quantum effects at such large length scales at very low temperatures, because otherwise the correlation length is just too small.