The arithmetic-to-spectrum conjecture for $L$-cuspidal lattices

Let $G_1$ and $G_2$ be semisimple algebraic groups over number fields $F_1$ and $F_2$, respectively. Let $K_i\subset G_i(\mathbb{A}_{F,f})$, $i=1,2$, be compact open subgroups, and let $\Gamma_{K_i}$ be the corresponding arithmetic lattices in $G_{i,\infty}$. Suppose that the adele groups $G_1(\mathbb{A})$ and $G_2(\mathbb{A})$ are isomorphic, and that the $L$-cuspidal spectrums of $(G_1(\mathbb{A}_{F_1}),K_1)$ and $(G_2(\mathbb{A}_{F_2}),K_2)$ are equal.

The arithmetic-to-spectrum conjecture. The lattices $\Gamma_{K_1}$ and $\Gamma_{K_2}$ are $L$-equivalent in $G_{1,\infty}\simeq G_{2,\infty}$.

This reverses the preceding spectrum-to-arithmetic direction and asks whether adelic arithmetic data determines the archimedean spectral multiplicities. The source calls the assertion obvious when the fields and groups coincide, but notes nontrivial examples and leaves the general claim open.