Fine deformation conjecture for self-dual Galois representations

Let $E$ be the number field in the deformation problem, let $V$ be an absolutely irreducible crystalline $p$-adic representation with the stated self-duality and distinct local data, and let $X_{V,f}$ be its fine deformation functor. Let $X_{V,\mathcal F}$ be the refined deformation functor associated with a refinement $\mathcal F$, and let $\partial\kappa$ be the induced weight-direction map. Fine and refined deformation conjectures.

1. $X_{V,f}$ is a closed point.
2. If $\mathcal F$ is noncritical, then $\partial\kappa$ is an isomorphism; in particular, $X_{V,\mathcal F}$ is formally smooth of dimension $m$.

These conjectures describe the expected rigidity of fine deformations and smoothness of the refined deformation space. The paper later relates them to the Bloch–Kato conjecture, but presents them here as conjectural.