Torsion local-global compatibility at $\ell=p$

Assume Conjecture 5.4. Let $\mathfrak{m}\subset \mathbf{T}^{T,\lambda,\tau}(K^p)$ be a non-Eisenstein maximal ideal. For $v\in S_p(F)$, set $\overline{\rho}_v:=\overline{\rho}_{\mathfrak{m}}|_{G_{F_v}}$ and $\rho_v:=\rho_{\mathfrak{m}}|_{G_{F_v}}$. Let $R^{\Box}_{\overline{\rho}_v}$ and $R^{\lambda_v,\preceq \tau_v}_{\overline{\rho}_v}$ be the corresponding framed and potentially crystalline deformation rings, let $\mathfrak{z}_{\lambda_v,\tau_v}^{\circ,\operatorname{int}}$ be the specified integral Bernstein-centre subring, and let $\eta$ denote the maps appearing in the diagram. Torsion local-global compatibility at $\ell=p$. For every $v\in S_p(F)$, there is a necessarily unique dotted arrow making the diagram commutative:

\begin{tikzcd} R^{\Box}_{\overline{\rho}_v} \arrow{r}{\rho_v} \arrow[two heads]{d}{} &\mathbf{T}^{T,\lambda,\tau}(K^p)_{\mathfrak{m}} \\ R^{\lambda_v,\preceq \tau_v}_{\overline{\rho}_v} \arrow[hook]{d}{} \arrow[dotted]{ur}{} &\mathfrak{z}_{\lambda_v,\tau_v}^{\circ,\operatorname{int}}\arrow{u}[swap]{\operatorname{nat}}\arrow{l}{\eta\mid_{\mathfrak{z}_{\lambda_v,\tau_v}^{\circ,\operatorname{int}}}}\arrow[hook]{d}{}\\ R^{\lambda_v,\preceq\tau_v}_{\overline{\rho}_v}[1/p]&\arrow{l}{\eta} \mathfrak{z}_{\Omega_v}. \end{tikzcd}

This conjecture is the proposed torsion local-global compatibility statement at places above $p$: the local Galois deformation and Bernstein-centre parameters should factor compatibly through the localized global Hecke algebra. Its status is not resolved in the supplied text.