Coefficient-field enlargement conjecture for compatible lisse sheaves

Let $X$, $E$, $\lambda$, and $\mathcal{E}$ be as in the compatibility conjecture for de Rham lisse sheaves, and let conditions (i) and (ii) denote respectively the coefficient condition and the de Rham condition appearing there. Coefficient-field conjecture. Condition (ii) implies condition (i) after replacing $E$ by a bigger number field contained in $E_\lambda$. This stronger statement is motivated by conjectures on Galois representations, but the supplied text gives no resolution status.