The cohomological spectrum-to-spectrum conjecture for arithmetic locally symmetric spaces

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 cohomological $L$-cuspidal spectrums of $(G_1(\mathbb{A}_{F_1}),K_1)$ and $(G_2(\mathbb{A}_{F_2}),K_2)$ are equal.

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

This conjecture extends the arithmetic-to-spectrum proposal by retaining only cohomological automorphic representations, the representations relevant to the cohomology and zeta functions of Shimura varieties. The source provides no general resolution.