  Let $K$ be a totally real number field of degree $r$. Let $K_{\infty}$ denote
the cyclotomic $\mathbb{Z}_{2}$-extension of $K$ and let $L_{\infty}$ be a
finite extension of $K_{\infty}$, abelian over $K$. The goal of this paper is
to compare the characteristic ideal of the $\chi$-quotient of the projective
limit of the narrow class groups to the $\chi$-quotient of the projective limit
of the $r$-th exterior power of totally positive units modulo a subgroup of
Rubin-Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters
$\chi$ of $\mathrm{Gal}(L_{\infty}/K_{\infty})$.
