The period-ratio conjecture for CM Hilbert cuspforms

Let $F$ be the CM field, let $F^+$ be its maximal totally real subfield, let $u$ be the Hecke character giving rise to the Hilbert cuspform $u(\theta)$, and let $C_{ u(\theta),cinfty}^{cepsilon}$, $cwidetilde{cOmega}_{cmathrm{CM},cinfty}$, and $cOmega_{cmathrm{CM},cinfty}$ be the complex periods occurring above. Write $cmathsf{t}$ for the parallel weight and let $ckappa_{cmu,1}^{cmax}$, $ckappa_{cmu,1}$, and $ckappa_{cmu,2}$ denote the associated infinity-type parameters. Fix an embedding $ciota_pöcolon coverline{cmathbb{Q}}cto coverline{cmathbb{Q}}_p$. Period-ratio conjecture. For every $cepsilon\text{ in }c{ cpm 1 c}^{I_{F^+}}$, the ratio

$$cfrac{cGamma((ckappa_{cmu,1}^{cmax}+1)cmathsf{t}-ckappa_{cmu,1})C_{cvartheta(ceta), cinfty}^{cepsilon}}{(-cwidetilde{cOmega}_{cmathrm{CM}, cinfty})^{ckappa_{cmu,2}}(-cOmega_{cmathrm{CM},cinfty})^{-ckappa_{cmu,1}}}$$

is a $p$-adic unit with respect to $ciota_p$. This conjecture is motivated by the expected equality, including $cmu$-invariants, of the two cyclotomic $p$-adic $L$-functions; it would imply that the cyclotomic Iwasawa main conjecture holds integrally rather than only after inverting $p$-power $cmu$-invariants.