The relative Frobenius algebraicity conjecture for abelian varieties

Let $A$ be an abelian variety over a number field $K$, let $L$ be the coefficient field used for its de Rham cohomology, and let $s\in \operatorname{End}(H^1_{\mathrm{dR}}(A,L))$. Relative Frobenius algebraicity conjecture. If $s$ is fixed by all but finitely many relative Frobenii, then $s$ is an $L$-linear combination of algebraic cycles. This is proposed as an analogue of Bost's theorem for general number fields, where relative Frobenii capture the information available from the $\ell$-adic monodromy group. The statement is presented as an expected input rather than as a result proved in the paper, and its general validity remains open.