Conjectural presentation of the tamely ramified Eisenstein module

Let $G$ be a reductive group satisfying the stated restriction that, for every root $\check{\alpha}\in\Lambda^{\vee}$, the map $\Lambda\to\mathbb{Z}$ given by $\lambda\mapsto\langle\check{\alpha},\lambda\rangle$ is surjective. Let $S=\{0,1,\infty\}$ be the ramification points, let $\mathcal{H}^{\otimes S}$ be the algebra generated by the Hecke operators at the points of $S$, and let $C_{Eis}$ be the module of tamely ramified Eisenstein series. For $\lambda\in\Lambda$, write $J_\lambda^s$ for the corresponding translation Hecke operator at $s\in S$, and for a simple reflection $s_\alpha\in W$, write $T_{s_\alpha}^s$ for the corresponding Hecke operator at $s$. Conjectural presentation. The module $C_{Eis}$ is generated as an $\mathcal{H}^{\otimes S}$-module by $\mathrm{Eis}_0$, subject to the relations

$$J_{\lambda}^0\mathrm{Eis}_0=J_{\lambda}^1\mathrm{Eis}_0=J_{\lambda}^{\infty}\mathrm{Eis}_0$$

for every $\lambda\in\Lambda$, and

$$(1+T_{s_\alpha}^0)(1+T_{s_\alpha}^1)\mathrm{Eis}_0=(1+T_{s_\alpha}^0)(1+T_{s_\alpha}^{\infty})\mathrm{Eis}_0=(1+T_{s_\alpha}^1)(1+T_{s_\alpha}^{\infty})\mathrm{Eis}_0$$

for every simple reflection $s_\alpha\in W$. This would give an explicit presentation of the Hecke module generated by the unramified Eisenstein series and describe the relations imposed by translations and simple reflections; the supplied text does not indicate that the conjecture has been proved or disproved.