Deligne–Ribet generation conjecture for Stark elements

Let $k$ be a totally real number field, let $k_\infty/k$ be its cyclotomic $\mathbb{Z}_p$-extension, let $L/k$ be the abelian extension and $\chi$ the character in the paper, and write $\mathbb{T}=T\otimes\Lambda$. Let $H^1(k_p,\mathbb{T})$ be the semi-local cohomology group at $p$, let $\mathbf{c}_{k_{\infty}}^{\textup{stark}}$ be the Stark element, and let $\textup{char}(\mathbb{A})$ denote the characteristic ideal of a torsion $\Lambda$-module $\mathbb{A}$. Let $\mathcal{L}_{k}^{\chi}$ be the Deligne–Ribet $p$-adic $L$-function attached to $\chi$. Deligne–Ribet generation conjecture. $\mathcal{L}_{k}^{\chi}$ generates

$$\textup{char}\left(\bigwedge^rH^1(k_p,\mathbb{T})/\Lambda\cdot\mathbf{c}_{k_{\infty}}^{\textup{stark}}\right).$$

For $k=\mathbb{Q}$ this specializes to the classical identification with the Kubota–Leopoldt $p$-adic $L$-function and hence to the main conjecture of Iwasawa theory; the assertion is proposed as its totally real, higher-rank analogue.