The R = T conjecture for Klingen-ordinary deformations

Let $\bar{\rho}$ be the residual Galois representation attached to a Klingen-ordinary automorphic representation, let $\mathbf{T}^{\mathrm{wKl}}_{\bar{\rho}}$ be the corresponding Hecke algebra, and let the universal weakly-Klingen-ordinary deformation ring represent the weak Klingen-ordinary deformation problem. A deformation valued in $\mathbf{T}^{\mathrm{wKl}}_{\bar{\rho}}$ is obtained from the Klingen-ordinary Hecke algebra. R = T conjecture for Klingen-ordinary deformations. This deformation is universal, so the universal weakly-Klingen-ordinary deformation ring is isomorphic to $\mathbf{T}^{\mathrm{wKl}}_{\bar{\rho}}$. The conjecture would identify the relevant deformation space with the Klingen-ordinary eigenvariety locally and is expected to imply the stated unobstructedness criterion.