The Frobenius compatibility conjecture for abelian varieties

Let $A$ be as above, put $L:=K_A^{\operatorname{conn}}$, and let $\operatorname{MT}_A^\sharp$ denote the set of conjugacy classes in $\operatorname{MT}_A$. For a nonzero prime ideal $\mathfrak p$ of $\mathcal O_L$ at which $A$ has good reduction, and each rational prime $\ell$ with $\mathfrak p\nmid\ell$, let $F_{\mathfrak p,\ell}$ be the conjugacy class of $\rho_{A,\ell}(\operatorname{Frob}_{\mathfrak p})$. Frobenius compatibility conjecture. There exists $F_{\mathfrak p}\in\operatorname{MT}_A^\sharp(\mathbb Q)$ such that $F_{\mathfrak p,\ell}=F_{\mathfrak p}$ for every such $\ell$. Equivalently, after every embedding $\iota\colon\mathbb Q_\ell\to\mathbb C$, the corresponding conjugacy class in $\operatorname{MT}_A(\mathbb C)$ is independent of $\ell$ and $\iota$. This conjecture expresses the expected compatibility of Frobenius conjugacy classes across the $\ell$-adic realizations; the supplied text gives no resolution.