Darmon–Vonk antisymmetry conjecture for Hecke-translated real quadratic singular moduli

Let $\tau,\omega\in\mathcal{H}_p$ be RM points with $\omega$ not in the $G$-orbit of $\tau$, let $\mathrm{T}\in\mathcal{I}_p$ be a Hecke operator, and let $J_p^\mathrm{T}(\tau,\omega)\in\mathbb{C}_p^\times$ be the associated value, defined up to multiplication by elements of $(\mathbb{C}_p^\times)_{\mathrm{tor}}\cdot\varepsilon_\tau^\mathbb{Z}$. Darmon–Vonk antisymmetry conjecture. One has

$$J_p^\mathrm{T}(\omega,\tau)=J_p^\mathrm{T}(\tau,\omega)^{-1}$$

up to multiplication by an element of $(\mathbb{C}_p^\times)_{\mathrm{tor}}\cdot\varepsilon_\tau^\mathbb{Z}\cdot\varepsilon_\omega^\mathbb{Z}$. This is the real-multiplication analogue of antisymmetry for differences of classical singular moduli, and the supplied text presents it as a conjecture motivated by numerical experiments; no proof or resolution is given.