On the structure of locally potentially equivalent Galois representations

Suppose $\rho_1, \rho_2$ are two $\ell$-adic Galois representations of the absolute group a number field, such that algebraic monodromy one is connected and locally potentially equivalent at set places positive upper density. We classify pairs show up to twisting by some representation, it given pair which trivial other abelian. Consequently, assuming first representation has group, we obtain equivalent, provided following conditions hold: (a) absolutely irreducible; (b) ranks groups equal; (c) second also (d) commutant image remains same upon restriction subgroups finite index group.