Coleman's finiteness conjecture for endomorphism algebras of abelian varieties

Let $F$ be a totally real field and let $g \in \mathbb Z_{\geq 1}$. For an abelian variety $X$ over $F$, write $\operatorname{End}(X)_{\mathbb Q}=\operatorname{End}(X)\otimes_{\mathbb Z}\mathbb Q$. Coleman's finiteness conjecture. The set

$$\{E=\operatorname{End}(X)_{\mathbb Q}: [E:\mathbb Q]=g \text{ and $X$ an abelian variety over $F$ with $\dim(X)=g$}\}$$

is finite. This conjecture predicts that only finitely many rational endomorphism algebras occur for $g$-dimensional abelian varieties over a fixed totally real field when the algebra has degree $g$.