Equivariant Fitting ideal conjecture for adjoint Hilbert modular L-values

Let $F$ be the totally real field, let $S$ be the finite set of places occurring in the construction, and let $E/F$ be a totally real abelian extension ramified away from $S$ or at $p$. Write $\Delta_E=\operatorname{Gal}(E/F)$, let $O$ be the coefficient ring, and let $\wp_{f_E}$ be the relevant congruence ideal for the base change $f_E$. The element ${\mathcal L}^{S}_{f,E}\in K[\Delta_E]$ is characterized by

$$\chi({\mathcal L}^{S}_{f,E})=\frac{G(\chi)^2\Gamma(\operatorname{ad}(\rho_f)\otimes\chi,1)L^{S_E}(\operatorname{ad}(\rho_f)\otimes\chi,1)}{\Omega_f^\Sigma\Omega_f^{\Sigma_F\backslash\Sigma}}$$

for every character $\chi\in\operatorname{Hom}(\Delta_E,\mathbf C^\times)$. Equivariant Fitting ideal conjecture. For each such extension $E/F$,

$${\mathcal L}^{S}_{f,E}\in \operatorname{Fitt}_{O[\Delta_E]}(\wp_{f_E}/\wp_{f_E}^2).$$

This conjecture gives an equivariant refinement of the expected relation between adjoint $L$-values and congruence ideals, and would imply the stated compatibility of the Euler-system construction with the un-primitive Coates--Schmidt $p$-adic $L$-function. Its status is not resolved in the supplied text.