Conjectural local-factor formula for singular moduli with relatively prime conductors

Let $d_1$ and $d_2$ be quadratic imaginary discriminants with relatively prime conductors. Write $f$ for the product of the two conductors, and for each prime $p$ let $d_{(p)}\in\{d_1,d_2\}$ be such that $p\nmid\operatorname{cond}(d_{(p)})$. For $x$ satisfying $x^2<d_1d_2$ and $x^2\equiv d_1d_2\pmod 4$, set $m_x=(d_1d_2-x^2)/4$; let $H$ be the correction term defined in the preceding valuation formula. Relatively prime conductor conjecture. For every prime $\ell$, the valuation $v_\ell(J(d_1,d_2)^2)$ is given by the displayed sum of $\epsilon_\ell(x)$ times the product of the stated local factors, with $\epsilon_\ell(x)$ as defined in the candidate statement. The formula is suggested by the local factor description for pairs of discriminants with relatively prime conductors; establishing it would extend the preceding valuation results beyond the cases proved in the paper.