Popescu's higher-rank abelian Stark conjecture

Let $K/k$ be a finite abelian extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $k$, let $E_S(K)$ be the group of $S_K$-units, let $E_S(K)^{\mathrm{ab}}$ be the subgroup of units $u$ such that $K(u^{1/w_K})/k$ is abelian, and let $r$ and $\eta'$ be as in Hypothesis StarkHhigher. Define

$$\Lambda_{K,S}^{\mathrm{ab}}=\left\{x\in\mathbb{Q}\bigwedge_{\mathbb{Z}[G]}^{r}E_S(K)\mid \varphi_1\wedge\cdots\wedge\varphi_{r-1}(x)\in\widetilde{E_S(K)^{\mathrm{ab}}}\text{ for all }\varphi_i\in E_S(K)^*\right\}.$$

Popescu's conjecture. Assuming Hypothesis StarkHhigher, one has

$$w_K\cdot\eta'\in\Lambda_{K,S}^{\mathrm{ab}}.$$

For $r=1$ this recovers Stark's abelian rank-one conjecture; in higher rank it predicts that the canonical exterior-power element built from leading $L$-values satisfies an arithmetic integrality and abelianity condition. The source presents it as an open conjecture and reports numerical verification in the studied cubic extensions.