The many-point Hecke-module presentation conjecture for tamely ramified Eisenstein series

Let $S\subset\mathbb{P}^1(\mathbb{F}_q)$ be a nonempty finite set of tame ramification points, let $C_{Eis}$ be the corresponding Eisenstein Hecke module, and let $\mathcal{H}^{\otimes S}$ be the tensor product of the local Hecke algebras. For $s\in S$, write $J_\lambda^s$ for the translation operator associated with $\lambda\in\Lambda$, and write $\mathrm{Avg}_{s_\alpha}^s$ for the averaging operator associated with a simple reflection $s_\alpha\in W$. Many-point presentation conjecture. If $\rho$ is integral, then $C_{Eis}$ is the $\mathcal{H}^{\otimes S}$-module generated by $\mathrm{Eis}_0$ subject to

$$(J_\lambda^p-J_\lambda^q)\mathrm{Eis}_0=0$$

for every $\lambda\in\Lambda$ and $p,q\in S$, and

$$\left(\prod_{s\in S\setminus\{p\}}\mathrm{Avg}_{s_\alpha}^s-\prod_{s\in S\setminus\{q\}}\mathrm{Avg}_{s_\alpha}^s\right)\mathrm{Eis}_0=0$$

for every simple reflection $s_\alpha\in W$ and $p,q\in S$. This is the stated natural generalization to several tame ramification points; the supplied text does not establish it or indicate that it has been resolved.