Leading term conjectures for Rubin–Stark elements

Let $L_n/K$, $S$, and $\Sigma$ be as in the source, let $r_T$ be the specified rank, and let $\varepsilon_{L_n\mid K,S,\Sigma}$ be the $r_T$-th Rubin–Stark element. Let $C_n^\bullet$ and $C_\infty^\bullet$ be the finite-level and Iwasawa-level complexes, with maps $\Theta_{L_n}$ and $\Theta$. Leading term conjectures. The following assertions should hold:

$$\varepsilon_{L_n\mid K,S,\Sigma}\in\bigcap\nolimits^{r_T}_{\mathds{Z}_p[\mathcal{G}_n]}U_{L_n,S,\Sigma}.$$

2. There is a basis $\mathfrak{z}_{L_n}\in\operatorname{Det}_{\mathds{Z}_p[\mathcal{G}_n]}(C_n^\bullet)$ such that

$$\Theta_{L_n}(\mathfrak{z}_{L_n})=\varepsilon_{L_n\mid K,S,\Sigma}.$$

3. There is a basis \mathfrak{z}_\infty\in\operatorname{Det}_{{\mathpalette% \raisebox{\depth}{\scalebox{1}[-1]{\mathsurround=0pt\relax\text{V}}}% }(C_\infty^\bullet) such that

$$\Theta(\mathfrak{z}_\infty)=(\varepsilon_{L_n\mid K,S,\Sigma})_n.$$

These are integrality and determinant-basis refinements of leading-term conjectures for equivariant $L$-functions and Rubin–Stark elements. The source presents the three assertions together and does not provide a resolution status.