I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the Selberg class is continuous. It seems that Selberg's orthogonality conjecture (SOC for short) implies such a continuity condition, judging by the following, which is a cross-post from MSE:

Indeed let $F$ be an element of the Selberg class and $\sigma$ a field automorphism of $\mathbb{C}$ that commutes with $F$. Let $\Phi_{\sigma}:F\mapsto \sigma\circ F\circ\sigma^{-1}$. Assuming $\Phi_{\sigma}$ maps a primitive element of the Selberg class to a primitive element of the Selberg class and Selberg's orthogonality conjecture, one gets that $\langle F,G\rangle=\langle \Phi_{\sigma}(F),\Phi_{\sigma}(G)\rangle$ where the inner product $\langle F,G\rangle$ is defined on the space $\mathcal{J}$ (which is defined as the complex linear space generated by the Selberg class) as the limit as $x$ tends to infinity of $\dfrac{1}{\log \log x}\sum_{p\le x}\dfrac{a_{p}(F)\overline{a_{p}(G)}}{p}$ with $a_{p}(F)$ such that for $\Re(s)>1$, $F(s)=\sum_{n>0}{\dfrac{a_{n}(F)}{n^s}}$. This makes $\Phi_{\sigma}$ continuous (more precisely, an automorphism of $\mathcal{J}$).

Now, writing $\langle F+\epsilon F,F+\epsilon F\rangle=\langle \Phi_{\sigma}(F+\epsilon F),\Phi_{\sigma}(F+\epsilon F)\rangle$, one gets $\langle \sigma\circ(F+\epsilon F)\circ\sigma^{-1},\sigma\circ(F+\epsilon F)\circ\sigma^{-1}\rangle=\langle F+\epsilon F, F+\epsilon F\rangle$, hence $\langle \sigma\circ F\circ\sigma^{-1}+\sigma\circ\epsilon F\circ\sigma^{-1},\sigma\circ F\circ\sigma^{-1}+\sigma\circ\epsilon F\circ\sigma^{-1}\rangle=\langle F+\epsilon F,F+\epsilon F\rangle$, thus $\langle\sigma(\epsilon)F,\sigma(\epsilon)F\rangle=\langle \epsilon F,\epsilon F\rangle$. Taking the limit of both sides of the equality as $\epsilon$ tends to $0$, one gets that $\lim_{\epsilon\to 0}\sigma(\epsilon)=0$, hence making $\sigma$ continuous.

My first question is: is this correct?

My second question is: is SOC already known to imply the Grand Riemann Hypothesis? If so, could I get some reference?

Thanks in advance.