Real Abelian Main Conjecture for arithmetic class groups

Let $\chi = \chi_0 \chi_p$, with $\chi_0 \ne 1$, be an even irreducible rational character and let $K=K_\chi$. For every irreducible $p$-adic character $\varphi$ dividing $\chi$, so that $\varphi_0$ divides $\chi_0$ and $\varphi_p=\chi_p$, write $({\mathcal E}_K / \widehat{\mathcal E}_K\,{\mathcal F}_K)_{\varphi_0}$ for the $\varphi_0$-component of the $\chi$-object ${\mathcal E}_K / \widehat{\mathcal E}_K\, {\mathcal F}_K$. Real Abelian Main Conjecture. One has

$$\# {\mathcal H}^{\rm ar}_{K,\varphi}=\#({\mathcal E}_K / \widehat{\mathcal E}_K\, {\mathcal F}_K)_{\varphi_0}.$$

This is the exceptional non-semisimple case of the Real Abelian Main Conjecture, relating the arithmetic class-group component to the corresponding quotient of units by the subgroup generated by proper-subfield units and Leopoldt's cyclotomic units. The supplied text states the conjecture but gives no evidence of its resolution.