Hida's simplicial concentration conjecture

Let $\Lambda$ be the Hida–Iwasawa algebra, let $R_h$ be the universal minimal ordinary deformation ring, let $\mathcal{R}_h$ be the associated simplicial deformation ring, and for an arithmetic weight $\lambda'$ let $P_{\lambda'}$ be the corresponding prime of $\Lambda$ and $\mathcal{R}_{\lambda'}$ the specialized simplicial deformation ring. Hida's simplicial concentration conjecture. The following equivalent statements should hold: $\mathrm{H}^{\bullet}_{\mathfrak m}$ is concentrated in degree $q_s$; and $\mathcal{R}_h$ is discrete, with

$$\operatorname{Tor}_\bullet^\Lambda(R_h,\Lambda/P_{\lambda'})\cong \pi_\bullet(\mathcal{R}_{\lambda'})$$

for every arithmetic weight $\lambda'$. This is the paper's conjecture of Concentration in Supremum Degree and is motivated by non-abelian Leopoldt conjectures; the stated equivalence follows from the preceding theorem.