Deformation-theoretic non-abelian Leopoldt conjecture

Let $R_h$ be the universal minimal ordinary deformation ring and let $\Lambda$ be the Hida–Iwasawa algebra. Deformation-theoretic non-abelian Leopoldt conjecture.

$$\dim R_h=\dim\Lambda-\ell_0.$$

This is identified in the paper as the deformation-theoretic non-abelian Leopoldt conjecture; the paper proves that it implies the concentration conjecture, but does not establish it in general.