The $p$-adic Beilinson conjecture for totally real fields

Let $K$ be a totally real field, let $S$ be the set of archimedean places together with the places above $p$, and let $\mathcal{L}$ and $\eta_{K,S}(j)$ be the generalized Iwasawa-theoretic element and generalized Stark element introduced in the paper. Let $\chi_{\rm cyc}$ denote the cyclotomic character.

The $p$-adic Beilinson conjecture. For each odd integer $j>1$,

$$\chi_{\rm cyc}^{1-j}(\mathcal{L})=\eta_{K,S}(j).$$

The paper uses this prediction to derive a generalized Coleman–Ihara formula; the supplied text gives no general proof or disproof.