Hida's codimension conjecture for nearly ordinary Hecke algebras

Let $F$) be a number field, let $S$ be the level data, let $\mathbf{h}^{n.ord}(S)$ be the nearly ordinary Hecke algebra, let $\Lambda$ be its weight algebra, and let $r_2$ denote the number of complex places of $F$. The support of $\mathbf{h}^{n.ord}(S)$ in $\Lambda$ is measured by its Krull dimension.

Hida's codimension conjecture. We have

$$\dim (\mathbf{h}^{n.ord}(S)\otimes \mathbb{Q}_p)=[F:\mathbb{Q}]+1,$$

so that the support of $\mathbf{h}^{n.ord}(S)$ in $\Lambda$ has codimension $r_2$.

This predicts the precise amount of torsion in the nearly ordinary Hecke algebra when $F$ has complex places. The surrounding discussion gives the corresponding totally real case and explains that the conjecture is open in the stated generality.