Chida's reciprocity conjecture for u_C

Let $[(r,\tau)]\in(\mathbb{Z}/f\mathbb{Z}\times H^{\mathcal{O}}(\mathfrak{N}))/\sim_f$ with $\tau$ reduced, set $\mathcal{O}=\mathcal{O}_{\tau}$, $\mathfrak{f}=f\mathcal{O}$, $\wp=p\mathcal{O}_K$, and $L=K(\mathfrak{f}\infty)^{\langle\sigma_{\wp}\rangle}$. Let $c\in C_K$ and write $\operatorname{rec}(c)=\sigma\in\operatorname{Gal}(K^{\mathrm{ab}}/K)$. Chida's reciprocity conjecture for $u_C$. The element $u_C(r,\tau)$ is a strong $p$-unit in $L$, and

$$u_{C,\delta}(r,\tau)^{\sigma^{-1}}=c\star u_{C,\delta}(r',\tau'),$$

where $\star$ is the defined action. This predicts both algebraicity and Shimura reciprocity for $u_C$; the source does not report a resolution.