Tate's correspondence conjecture for isomorphic two-dimensional Galois representations

Let $X_1$ and $X_2$ be varieties defined over $\mathbb{Q}$, and let $\rho_1$ and $\rho_2$ be isomorphic two-dimensional Galois representations occurring in their étale cohomology. A correspondence between $X_1$ and $X_2$ is an algebraic cycle on $X_1\times X_2$; it is defined over $\mathbb{Q}$ when the cycle is defined over $\mathbb{Q}$. The Tate conjecture. If $\rho_1$ and $\rho_2$ are isomorphic, there should be a correspondence between $X_1$ and $X_2$ defined over $\mathbb{Q}$ that induces an isomorphism between $\rho_1$ and $\rho_2$. This predicts that isomorphisms between Galois-representation pieces of the étale cohomology of varieties have a geometric realization by algebraic cycles defined over the ground field. The source formulates the claim as the Tate conjecture and gives no resolution evidence here.