Serre's conjecture on the component-group image of algebraic monodromy

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. For an algebraic group $G$, write $\pi_0(G)$ for its group of connected components. Serre's conjecture. The natural map

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

is an epimorphism. The assertion says that the algebraic Galois monodromy group meets every connected component of the algebraic Sato–Tate group. The source attributes this conjecture to Jean-Pierre Serre and gives no evidence that it has been resolved.