Gross's conjecture on Fourier coefficients of quaternionic modular forms

Let $G$ be the group supporting quaternionic modular forms, and let $\mathbb X$ be the representation of its Levi subgroup $M\cong\operatorname{GL}_2$ identified with $\operatorname{Sym}^3\otimes\det^{-1}$. Let $f$ be a holomorphic modular cusp form of even weight $2k$ with level $1$ and trivial character, and let $\mathcal F$ be the associated cuspidal quaternionic modular form of weight $k$. For $\mathcal E\in\mathbb X(\mathbb Z)$ corresponding to the ring of integers of a totally real étale cubic algebra $E/\mathbb Q$, write $a_{\mathcal E}(\mathcal F)$ for the Fourier coefficient indexed by $\mathcal E$. Let $V_E$ be the $2$-dimensional Artin representation defined by

$$\operatorname{Ind}_E^\mathbb Q\mathbbm{1}=\mathbbm{1}\oplus V_E,$$

and let $\Delta_E$ be the discriminant of $E$. Gross's conjecture. For every such $\mathcal E$, one has

$$a_{\mathcal E}(\mathcal F)^2=L\left(\frac12,f\otimes V_E\right)\,\Delta_E^{k-\frac12}.$$

This is an analogue of Waldspurger's theorem relating Fourier coefficients to special $L$-values, and would describe the arithmetic content of cuspidal quaternionic modular forms associated with $\mathrm{PGL}_2$ forms. The conjecture is presented as an open conjecture in the source.