The freeness conjecture for the tamely ramified Eisenstein Hecke module

Let $W$ be the Weyl group, let $\Lambda$ be the coweight lattice, and let $\widetilde{C}$ be the Hecke module generated by the relevant tamely ramified Eisenstein series. Freeness conjecture. $\widetilde{C}$ is free of rank $|W|^2$ over $\mathbb{C}[\Lambda]$. This is presented as a conjectural structural property of the Hecke module; the supplied text gives no resolution status or proof of the assertion.