The quaternionic Maass Spezialschar Dirichlet-series conjecture for split $\mathrm{SO}(8)$

Let $\varphi$ be a level-one cuspidal quaternionic modular form, let $\Pi$ be the irreducible automorphic representation generated by $\varphi$, and let $[T_1,T_2]\in\mathrm{M}_2({\mathbf Z})^{\oplus 2}$ be strongly primitive with $T=S([T_1,T_2])$. Let $S$ be a finite set of primes containing all primes dividing $8\det(T)$, let $\mathrm{M}_2^S({\mathbf Z})$ denote the integer $2\times2$ matrices with nonzero determinant whose determinants are divisible only by primes outside $S$, and let $E$ be the quadratic field determined by the discriminant of $T$. Write $a_\varphi([T_1,T_2])$ for the corresponding Fourier coefficient, and write $L^S(\Pi,\operatorname{Std},s)$ and $\zeta_E^S(s)$ for the partial standard $L$-function and partial Dedekind zeta function. Dirichlet-series conjecture. One has

$$a_\varphi([T_1,T_2])\frac{L^S(\Pi,\operatorname{Std},s)}{\zeta^S(2s)\zeta_E^S(s+1)}=\sum_{g\in\mathrm{M}_2^S({\mathbf Z})/\operatorname{GL}_2({\mathbf Z})}\frac{a_\varphi([T_1,T_2]\cdot g)}{|\det(g)|^{s+\ell-1}}.$$

This conjecturally extends Andrianov's Dirichlet-series formula from Siegel modular forms on $\operatorname{Sp}_4$ to quaternionic modular forms on split $\operatorname{SO}(8)$, relating Fourier coefficients to the standard $L$-function of $\Pi$ and the Dedekind zeta function of the quadratic field attached to the discriminant. Its resolution is not specified in the supplied text.