Iwasawa theory of Rubin-Stark units and class group

Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. In this article, we extend results of \cite{Kazim109} relating characteristic ideal of the $\chi$-quotient of the projective limit of the ideal class groups to the $\chi$-quotient of the projective limit of the $r$-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some $\overline{\mathbb{Q}_{p}}$-irreductible characters $\chi$ of $\mathrm{Gal}(L_{\infty}/K_{\infty})$.