Weak algebraic Sato–Tate conjecture

Let $A$ be an abelian variety over $K$, let $\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}$ be its normalized algebraic $\ell$-adic monodromy group, and let $\operatorname{AST}_K(A)$ be the algebraic Sato–Tate group. Write $\pi_0(G)$ for the group of connected components of an algebraic group $G$, and let

$$d:\pi_0\bigl(\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}\bigr)\longrightarrow\pi_0\bigl(\operatorname{AST}_K(A)\bigr)$$

be the natural map. Weak algebraic Sato–Tate conjecture. The map $d$ is an isomorphism. This strengthens Serre's component-group epimorphism by asserting that the two component groups coincide, but the supplied text does not state whether it is resolved.