I define L-rigs in Are there infinitely many L-rigs?

An interesting property of such classes of L-functions is that if any two elements thereof fulfill RH, then so does their product. On the other hand, the Ramanujan conjecture (which is assumed as elements of an L-rig are required to belong in the Selberg class) is stable under the Rankin-Selberg convolution.

A reformulation of RH is that any non-trivial zero $s$ of an L-function $F$, where here by L-function I mean any element of the maximal L-rig, is invariant under the involution $s\mapsto 1-\bar{s}$. Analogously, a reformulation of RC is that any Satake parameter at an unramified prime is invariant under the involution $\alpha\mapsto\bar{\alpha}^{-1}$.

Another common point is that the Satake parameters at $p$ are the reciprocal roots of the local factor $F_{p}$ of $F$, while non trivial zeros of $F$ are roots of its complete L-function. Call then the non trivial zeros of $F$ "$\times$-essential roots" of $F$ and its Satake parameters at unramified primes "$\otimes$-essential roots" of $F$.

Now, viewing the permutation groups $G_{\times}(F)$ of the $\times$-essential roots of $F$ and $G_{\otimes}(F)$ of the $\otimes$-essential roots of $F$ preserving it and compatible with the structure of L-rig as Galois groups, are those two Galois groups isomorphic? In other words, in an L-rig, is the Ramanujan conjecture equivalent to the Riemann hypothesis?