Number theoretic phrasing
Let $G$ be a connected reductive group over a characteristic $0$ field $F$. Associated to $G$ is its Langlands dual group $^{L}G$. For every dominant cocharacter $\mu$ of $G_{\overline{F}}$ Kottwitz constructs in [Kot1,(2.1.1)] a representation $r_{-\mu}:\, ^{L}(G_E)\to \mathrm{GL}(V)$ where $E$ is the reflex field of $\mu$. Given two $L$-parameters $\psi_1,\psi_2:W_F\to {^L}G$ I believe that the equivalence $r_{-\mu}\circ \psi_1\sim r_{-\mu}\circ\psi_2$ for all $\mu$ does not imply the equivalence of $\psi_1$ and $\psi_2$ (e.g. this is true for some $G$ but examples for Spin groups are given in [Lar]). I am under the impression that if one considers not just a single $\mu$ but tuples of $\mu$'s simultaneously that this fixes the issue. I think this is supposed to be pivotal to the philosophy of why V. Lafforgue and his predecessors consider moduli spaces of Shtukas with multiple legs (but I am really interested in the result I asked--not how Lafforgue deals with it which I understand is slightly different).
Can anyone provide a precise statement? Is it that if for all tuples $(\mu_1,\ldots,\mu_n)$ one has that the representations $r_{-\mu_i}\circ\psi_1$ and $r_{-\mu_i}\circ\psi_2$ are simultaneously isomorphic (in some sense?) then $\psi_1\sim\psi_2$?
Lie group theoretic statement
Suppose that $G$ is a complex reductive Lie group and that $\psi_1,\psi_2:\Gamma\to G$ are two continuous homomorphisms where $\Gamma$ is some topological group (in my setting it's something like a Galois group, but not compact). I believe it's known that if $\rho\circ\psi_1$ is conjugate to $\rho\circ\psi_2$ for all irreducible algebraic representations $\rho$ of $G$ then it needn't be true that $\psi_1$ and $\psi_2$ are $G$-conjugate (cf. again the paper [Lar]). I believe that one can remedy this if one considers tuples of representations.
Can anyone provide a precise statement? Is it true that if for all tuples $(\rho_1,\ldots,\rho_n)$ one has that the representations $\rho_i\circ\psi_1$ and $\rho_i\circ\psi_2$ are simultaneously isomorphic (in some sense?) then $\psi_1$ and $\psi_2$ are $G$-conjugate?
[Lar] Larsen, M., 1994. On the conjugacy of element-conjugate homomorphisms. Israel Journal of Mathematics, 88(1-3), pp.253-277.
[Kot1] Kottwitz, R.E., 1984. Shimura varieties and twisted orbital integrals. Mathematische Annalen, 269(3), pp.287-300.
