The syntomic regulator formula for discrete valuation subfields

Let $K\subset\mathbb C_p$ be a discrete valuation subfield, meaning that its valuation is induced from the valuation on $\mathbb C_p$, and let $R$ be the valuation ring of $K$. Assume the Beilinson–Soulé conjecture holds in characteristic zero when $n\geq 3$. Let $\widetilde{\mathcal M}_n(K)$ be the complex whose cohomology maps to $K_{2n-1}^{(n)}(K)$, and identify $K_{2n-1}^{(n)}(K)\cong K_{2n-1}^{(n)}(R)$. Regulator formula conjecture. For all $n\geq 2$, the regulator map

$$H^1(\widetilde{\mathcal M}_n(K))\longrightarrow K_{2n-1}^{(n)}(K)\cong K_{2n-1}^{(n)}(R)\longrightarrow K$$

is given by the same formula as before: it maps $\{x\}_n$ to $\pm (n-1)!\,\operatorname{Li}_n(x)$. This would extend the explicit cyclotomic regulator formula to arbitrary discrete valuation subfields of $\mathbb C_p$; the source states the claim conditionally on the Beilinson–Soulé conjecture but gives no evidence of a proof or disproof.