The motivic-weight-22 classification conjecture for geometric Galois representations

Let $V_f$ be the geometric Galois representation associated with a Hecke eigenform, and let $\phi_1,\ldots,\phi_{11}$ be the representations associated with the $11$ automorphic representations in Theorem 1. Let $\mathbb L$ denote the Tate Galois representation, and let a representation have non-negative Hodge--Tate weights and motivic weight at most $22$. Classification conjecture. Every semisimple geometric Galois representation of motivic weight at most $22$, and with non-negative Hodge--Tate weights, is a direct sum of representations $\phi_j$ for $1\leq j \leq 11$ tensored with $\mathbb L^k$ for some $k \geq 0$. This is presented as part of the Langlands-program framework and is not established in the source; it would classify the relevant semisimple geometric Galois representations in terms of the listed automorphic ones.