Hida's dimension conjecture for ordinary completed cohomology

Let $F$ be the number field, $p$ the fixed prime, $U$ a product subgroup as above, $S$ a finite set of finite places containing those dividing $p$, and $\Lambda$ the ordinary weight Iwasawa algebra. Let ${\mathbb T}^{S}_\text{ord}(U)$ be the ordinary Hecke algebra acting faithfully on $H^\ast_\text{ord}(U)$, and let ${\mathfrak m}$ be a non-Eisenstein maximal ideal, meaning that the associated residual Galois representation is absolutely irreducible. Hida's conjecture.

$$\dim_\Lambda H^\ast_\text{ord}(U)_{\mathfrak m}=\dim\Lambda-l_0.$$

This predicts the expected codimension of localized ordinary completed cohomology over weight space and is the dimension formula underlying Hida's conjectural relationship between ordinary Hecke algebras and Galois deformation rings. The source gives no resolution status.