Darmon–Vonk conjecture on prime exponents of real quadratic singular moduli

Let $D_1,D_2$ be coprime fundamental discriminants, let $\tau_1,\tau_2$ be the corresponding real quadratic points, and let $p\leq 13$ be prime. Let $J_p(\tau_1,\tau_2)$ be the Darmon–Vonk $p$-adic invariant, and let $\mathfrak{q}$ lie above an integer prime $q\neq p$. If $q$ is split in $\mathbb{Q}(\sqrt{D_1})$ or $\mathbb{Q}(\sqrt{D_2})$, then $\operatorname{ord}_{\mathfrak{q}}(J_p(\tau_1,\tau_2))=0$. Otherwise, let ${\mathrm{O}}$ be a maximal order in the quaternion algebra ramified at $p,q$. Darmon–Vonk's conjecture. There exist optimal embeddings $\phi_1,\phi_2$ of discriminants $D_1,D_2$ into ${\mathrm{O}}$ such that

$$\operatorname{ord}_{\mathfrak{q}}(J_p(\tau_1,\tau_2))=\operatorname{Int}_{{\mathrm{O}}}^q(\phi_1,\phi_2).$$

In other words, the exponents of primes above $q$ in the factorization of $J_p(\tau_1,\tau_2)$ are given by $q$-weighted intersection numbers associated to optimal embeddings of $D_1,D_2$ into a maximal order in the indefinite quaternion algebra ramified at $p,q$. The conjecture proposes an arithmetic interpretation of the prime exponents of the real quadratic analogue of $j(\tau_1)-j(\tau_2)$; the source provides no evidence of a resolution.