The Algebraic Sato--Tate conjecture for abelian varieties

Let $A$ be an abelian variety of dimension $g$ over a number field, and let $G_\ell^{1,\rm zar}\subseteq \operatorname{Sp}_{2g}$ be the Zariski closure of the determinant-one part of its $d9\ell$-adic Galois representation. Algebraic Sato--Tate conjecture. There exists a $\mathbf{Q}$-algebraic subgroup $\operatorname{AST}(A)$ of $\operatorname{Sp}_{2g}$ such that

$$G_\ell^{1,\rm zar}=\operatorname{AST}(A)\otimes_{\mathbf{Q}}\mathbf{Q}_\ell.$$

The conjecture seeks a rational algebraic group governing the full, including possibly disconnected, $d9\ell$-adic monodromy group. The source presents it as closely related to the Mumford--Tate conjecture and gives no general resolution.