In fact, I have quite good reasons to think that the above conjecture (the only automorphisms of $\mathcal{S}$ are the identity and the complex conjugation) is equivalent to the Riemann Hypothesis for the whole Selberg class. Certainly one would wish for a stricter proof here... :-)
EDIT: Considering David Hansen's answer, it appears that the group of automorphisms of $\mathcal{S}$ might be richer than I first expected. In fact I may have been a bit hasty saying "is equivalent to". An interesting question would be to ask whether the action of this group on $\mathcal{S}$ is transitive or not.