Gross's conjecture: the dihedral case

Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic modular forms on $\mathsf{G}_2$ associated via functoriality with certain modular forms on $\mathrm{PGL}_2$, Gross conjectured in 2000 that their Fourier coefficients encode $L$-values of cubic twists of the modular form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). We prove Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known.