Serre's algebraic Sato–Tate conjecture

Let $A$ be an abelian variety with rational symplectic representation $(V,\psi)$, and let $K$ be a field over which $A$ is defined. Write $\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}$ for the algebraic $\ell$-adic monodromy group of $A$ with determinant normalized to one. Algebraic Sato–Tate conjecture. There is a reductive algebraic group $\operatorname{AST}_K(A)$ defined over $\mathbb Q$, natural in $K$, with

$$\operatorname{AST}_K(A)\subseteq \operatorname{Sp}_{(V,\psi)},$$

and, for every prime number $\ell$, a natural-in-$K$ monomorphism of group schemes

$$\operatorname{ast}_{\ell,K}:\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}\hookrightarrow \operatorname{AST}_K(A)_{\mathbb Q_\ell}.$$

Moreover, this monomorphism is an isomorphism:

$$\operatorname{ast}_{\ell,K}:\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}\xrightarrow{\simeq}\operatorname{AST}_K(A)_{\mathbb Q_\ell}.$$

The conjecture proposes a motivic algebraic group over $\mathbb Q$ whose base change to $\mathbb Q_\ell$ captures the normalized algebraic Galois monodromy group. The supplied text attributes its development to Serre, Banaszak and Kedlaya, but gives no resolution status.