R=T and étaleness conjectures for refined deformation rings

Let $R_{\rho,\mathcal F}$ be the ring pro-representing the refined deformation functor, let $\mathbb T$ be the completed local Hecke algebra at the corresponding eigenvariety point $z$, and let $\kappa$ be the weight map. R=T and criticality conjectures.

1. The natural map

$$R_{\rho,\mathcal F}\longrightarrow\mathbb T$$

is an isomorphism. 2. The map $\kappa^{\sharp}$ is an isomorphism; equivalently, the weight map $\kappa$ is étale at $z$.

These assertions are the eigenvariety and deformation-theoretic consequences expected from the preceding conjectures. They are presented as conjectural in the source and remain open in general.