Darmon–Dasgupta's real quadratic elliptic unit conjecture

Let $p$ be a prime, let $\tau\in\mathcal{H}_p$ be a real quadratic point of discriminant $D$ prime to $p$, and let $H_\tau$ be the narrow ring class field of the order defined by $\tau$. Let $J_{\mathrm{DR}}[\tau]$ denote the real quadratic elliptic unit attached to $\tau$. Darmon–Dasgupta's conjecture. One has

$$J_{\mathrm{DR}}[\tau]\in\mathcal{O}_{H_\tau}[1/p]^{\times}.$$

This predicts that the values are $p$-units in the relevant narrow ring class fields, making them analogues of classical elliptic units.