Finite flat complete-intersection conjecture for deformation rings

Let $S$ be the fixed set of primes and let $S'=s'_1,\dots,s'_n\}$ be a ramified auxiliary set whose existence is guaranteed by Theorem 1. The ring $R_{S\cup S'}$ is a finite flat complete intersection over the Witt-vector ring $W({\bf k})$. Finite flat deformation-ring conjecture. The ring

$$R_{S\cup S'}$$

is a finite flat complete intersection over $W({\bf k})$. This conjecture concerns the finer structure of deformation rings. The source notes that work of Taylor, using arguments of G. Böckle, implies it for odd $\overline{\rho}$ in many cases, while there is some evidence in the even case.