Ribet–Ihara-type conjecture for cohomology of Shimura curves

Let $S$ be a finite set of primes not dividing $M\ell$. Fix $f\in S_2(\Gamma_0(N\Delta'\ell^2S),\psi)$ that is supercuspidal of type $\tau$ at $\ell$ and special at primes $p\mid\Delta'$. Let $\mathcal R_S$ be the universal Galois deformation ring and let ${\bf T}^\psi_S$ be the Hecke algebra acting on the corresponding newforms. For the Shimura curve

$${\bf X}_1(NS)=B^\times\setminus B_{\bf A}^{\times}/K_\infty^+V_1(NS),$$

with

$$V_1(NS)=\prod_{p\nmid NS\ell}R_p^\times\prod_{p\mid NS}K_p^1(N)\times(1+u_\ell R_\ell),$$

let $\mathcal M^\psi_S=H^1({\bf X}_1(NS),\mathcal O)_{\mathfrak m_S}^{\widehat\psi}$, where $K_p^1(N)$ is as defined in the source and $u_\ell$ is a uniformizer of $B_\ell^\times$. The conjecture. The map

$$\mathcal R_S\to{\bf T}^\psi_S$$

is an isomorphism of complete intersections, and $\mathcal M^\psi_S$ is a free ${\bf T}^\psi_S$-module of rank $2$. This extends the expected minimal-level deformation-theoretic and cohomological results to levels with ramification allowed at the primes in $S$; it is motivated by the absence of a quaternionic analogue of Ihara's lemma at nonminimal level and is presented as an open conjecture.