The local factor formula for the arithmetic intersection number of quadratic imaginary orders

Let $d_1$ and $d_2$ be discriminants of quadratic imaginary orders, and fix an integer $t$. Let

$$m:= \frac14\bigl(d_1d_2-(d_1d_2-2t)^2\bigr).$$

Assume that the conductors of $d_1$ and $d_2$, together with $m$, have no simultaneous common factor. For each prime $p$, let $d_{(p)}\in\{d_1,d_2\}$ be such that the quadratic imaginary order of discriminant $d_{(p)}$ is maximal at $p$, and let $f_1$ denote the conductor of $d_1$. The local factor formula conjecture.

$$\mathscr{J}(d_1,d_2,t)=\prod_{p\mid m,\,p\ne\ell}\begin{cases} 1+v_p(m) & \left(\frac{d_{(p)}}{p}\right)=1,\ p\nmid f_1,\\ 2 & \left(\frac{d_{(p)}}{p}\right)=1,\ p\mid f_1,\ \textup{or}\\ & p\mid d_{(p)},\ (d_{(p)},-m)_p=1,\ p\nmid f_1\\ 1 & \left(\frac{d_{(p)}}{p}\right)=-1,\ p\nmid f_1,\ v_p(m)\textup{ even}\ \textup{or}\\ & p\mid d_{(p)},\ (d_{(p)},-m)_p=1,\ p\mid f_1,\ v_p(m)=2\\ 0 & \textup{otherwise}.\end{cases}$$

The conjecture gives a product of local factors for the arithmetic intersection quantity $\mathscr{J}(d_1,d_2,t)$ in the stated coprimality case.