Dimension conjecture for the ordinary deformation ring

Let $R_\rho^{\mathrm{ord}}$ be the ordinary deformation ring attached to the Galois representation $\rho$ and let $\Lambda_E$ be the ordinary weight algebra, with $r=\operatorname{rk}\mathrm T$ and $l_0$ the defect of $\mathrm G$. Dimension conjecture for the ordinary deformation ring. The ring has dimension

$$\dim R_\rho^{\mathrm{ord}}=\dim\Lambda_E-l_0=r-l_0.$$

This is the Galois-deformation counterpart of the dimension conjecture for the ordinary cohomology complex. It is used to make the derived Hecke action factor through the dual of a dual Selmer group, and remains conditional in the generality considered.