The Taylor–Wiles-level Galois lifting conjecture

Let $\overline{\rho}_{\mathfrak{m}}$ be the residual representation from the preceding conjecture, and let $(Q,(\gamma_{v,1},\dots,\gamma_{v,n})_{v\in Q})$ be a Taylor–Wiles datum. Let $G_{F,S\cup Q}$ be the Galois group unramified outside $S\cup Q$, and let ${\mathbb T}^{S\cup Q}(U^p_1(Q))_{\mathfrak{m}_{Q,1}}$ be the localized Taylor–Wiles-level Hecke algebra.

Taylor–Wiles-level lifting conjecture. If $\overline{\rho}_{\mathfrak{m}}$ is absolutely irreducible, there exists a continuous lifting

$$\rho_{\mathfrak{m},Q}:G_{F,S\cup Q}\longrightarrow\operatorname{GL}_n({\mathbb T}^{S\cup Q}(U^p_1(Q))_{\mathfrak{m}_{Q,1}})$$

whose Frobenius characteristic polynomial at every finite place $v\notin S\cup Q$ is

$$X^n-T_v^1X^{n-1}+\cdots+(-1)^iq_v^{i(i-1)/2}T_v^iX^{n-i}+\cdots+(-1)^nq_v^{n(n-1)/2}T_v^n,$$

and which is of type ${\mathcal S}_Q$.

This is the refinement of the global Galois representation conjecture required at Taylor–Wiles level. The paper assumes it and does not provide a resolution.