Soulé–Stark conjecture for rank-one motives

Let $(M/K,R)$ be a rank-one motive, let $T$ be its $p$-adic realization, and let $r=r(M/K,R)$. Under the source's hypotheses, the rank-$r$ Soulé twist sends the inverse system of Rubin–Stark elements to the Soulé–Stark element

$$\beta_S^b(T):={\rm tw}_T^r\left(\varprojlim_n\eta_S^w(\varepsilon\cdot\mathds{Z}_{p,L_n/K})\right)\in {\bigcap}_{\mathcal{R}_p}^rH^1(\mathcal{O}_{K,S},T^\ast(1)).$$

Soulé–Stark conjecture. The Soulé–Stark element coincides with the Bloch–Kato element:

$$\eta_S^b(T)=\beta_S^b(T).$$

This conjecture proposes that the $p$-adic family of Rubin–Stark elements recovers the Bloch–Kato special element for every rank-one motive. It is proved in several special cases described later in the paper, but remains open in general.