Non-abelian Leopoldt dimension conjecture for ordinary cohomology

Let $\Lambda_E$ be the regular local ordinary weight algebra of dimension $r=\operatorname{rk}\mathrm T$, let $F_{\mathfrak m}^{\bullet}$ be the complex interpolating the localized ordinary cohomology, and let $l_0$ be the defect of $\mathrm G$. Non-abelian Leopoldt dimension conjecture. The cohomology of this complex has Krull dimension

$$\dim_{\Lambda_E}H^*(F_{\mathfrak m}^{\bullet})=r-l_0.$$

This is presented as a higher, non-abelian analogue of Leopoldt's conjecture and is related to a separate Galois-theoretic dimension conjecture. Its validity is used to control the derived Hecke action.