Integral Hecke stability conjecture for coherent cohomology lattices

Let $\kappa$ be a dominant weight for $M_\mu/Z_s(G)$, let $V_\kappa$ be the corresponding Weyl representation over $\mathcal{O}_{E',\lambda'}$, and let $\mathcal{H}^{\int}_{p,\kappa,\iota}$ be the integral Hecke algebra acting on the cohomology. Denote by $\mathrm{H}^i(K,V_\kappa)_{\overline{\mathbb{Z}}_p}$ and $\mathrm{H}^i_{cusp}(K,V_\kappa)_{\overline{\mathbb{Z}}_p}$ the associated cohomology lattices. Integral Hecke stability conjecture. The lattices $\mathrm{H}^i(K,V_\kappa)_{\overline{\mathbb{Z}}_p}$ and $\mathrm{H}^i_{cusp}(K,V_\kappa)_{\overline{\mathbb{Z}}_p}$ are stable under $\mathcal{H}^{\int}_{p,\kappa,\iota}$. This asserts integral stability of both ordinary and cuspidal coherent cohomology under the integral Hecke algebra; the source gives no general proof or resolution.