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Fix a prime $ p $. Let $ \mathbb{F} $ be a finite field of characteristic $ p $ and $ W(\mathbb{F}) $ the ring of Witt vectors of $\mathbb{F} $ . Let $ \widehat{\mathfrak{A}}_{W(\mathbb{F})} $ be the category of complete Noetherian local $ W(\mathbb{F}) $-algebras with residue field $ \mathbb{F} $. Let $ K $ be a number field, $ S $ a finite set of primes of $ K $ containing any primes above $ p $ and $ \infty $, and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ unramified outside $ S $.

Mazur's dimension conjecture on universal deformation ring states the following, cf. Section 1.10 in [Mazur, B. (1989). Deforming galois representations. In Galois Groups over $ \mathbb{Q} $ (pp. 385-437). Springer, New York, NY.]

Conjecture:(Mazur) Let $ \bar{\rho}:G_{K,S}\to {\rm GL}_{n}(\mathbb{F}) $ be an absolutely irreducible representation of $ G_{K,S} $, and $ R^{\text{univ}} $ the universal deformation ring, then we have $$\text{Krulldim}(R^{\text{univ}}/(p))= h^{1}(G_{K,S},\text{ad}~\bar{\rho})-h^{2}(G_{K,S},\text{ad}~\bar{\rho}), $$ where $ h^{i}:=\text{dim}_{\mathbb{F}}~H^{i}(G_{K,S},\text{ad}\bar{\rho}) $ for $ i=1,2 $ and $ \text{ad}(\bar{\rho}) $ is the adjoint representation of $ \bar{\rho} $.

Recall that a deformation of $ \bar{\rho} $ to $ A\in \widehat{\mathfrak{A}}_{W(\mathbb{F})} $ is an equivalence class of lifting and the universal deformation ring $ R^{\text{univ}} $ always exists in $ \widehat{\mathfrak{A}}_{W(\mathbb{F})} $.

Question: What is the current status of this conjecture? What results are known in its direction?

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