Pseudorepresentation compatibility conjecture for patched completed homology

Let $R^{\operatorname{ps}}_p$ be the relevant pseudodeformation ring, let $E_{\mathfrak{B}}$ be the corresponding Hecke-module coefficient algebra, and let $\widetilde{\mathcal{C}}(\infty)$ be the patched completed-homology complex. The ring $R^{\operatorname{ps}}_p$ acts on $H_{q_0}(\widetilde{\mathcal{C}}(\infty))$ in two ways: through the map $R^{\operatorname{ps}}_p\to E_{\mathfrak{B}}$ and through the map $R^{\operatorname{ps}}_p\to\mathbb{T}^S(U^p)_{\mathfrak{m}}$. Pseudorepresentation compatibility conjecture. These two actions of $R^{\operatorname{ps}}_p$ on $H_{q_0}(\widetilde{\mathcal{C}}(\infty))$ coincide. This compatibility is an additional assumption used to make the depth-estimate part of the Taylor–Wiles method work for the relevant images in the category of $E_{\mathfrak{B}}$-modules; its resolution is not specified in the supplied text.