The integrality conjecture for the regulator element ${\mathfrak S}_{K/k}$

Let $k$ be a totally real number field, let $p$ be an odd prime splitting completely in $k$, and let $K$ be as in the paper. Write $G=\operatorname{Gal}(K/k)$ and let $\delta(K/k)$ be the exponent appearing in the paper's integrality bound. The integrality conjecture. One has

$${\mathfrak S}_{K/k}\subset p^{\delta(K/k)}{\mathbb Z}_pG$$

for every such $K$. The claim strengthens the integrality consequences proved earlier for the twisted zeta-function constructions. The source gives no resolution, so the conjecture remains open.