Equivariant adjoint L-value congruence conjecture for Hilbert modular forms

Let $F$ be the totally real field, let $S$ be the set of places dividing the level and $p$, and let $E/F$ be a totally real abelian extension ramified away from $S$ or at $p$. Set $A_E=O[\operatorname{Gal}(E/F)]$, let $\wp_{f_E}$ be the relevant congruence ideal, and let ${\mathcal L}^{\operatorname{an}}_E(\operatorname{ad}(\rho_f))$ be the equivariant adjoint $L$-value formed from the twisted special values and character idempotents. Equivariant adjoint L-value congruence conjecture. For every such $E/F$:

$${\mathcal L}^{\operatorname{an}}_E(\operatorname{ad}(\rho_f))\in A_E,$$

${\mathcal L}^{\operatorname{an}}_E(\operatorname{ad}(\rho_f))$ annihilates $\wp_{f_E}/\wp_{f_E}^2$, belongs to the Fitting ideal of $\wp_{f_E}/\wp_{f_E}^2$ over $A_E$, and generates that Fitting ideal over $A_E$. The four assertions are presented together as an equivariant formulation of the congruence number formula established by Hida; the supplied text does not state that they are resolved.