The prime-divisor conjecture for twisted Borcherds product CM values

Let $F$ be the real quadratic field and let $K$ be a CM extension of $F$ as in the preceding construction. Write $\mathcal{O}_F$ for the ring of integers of $F$, let $\mathfrak l$ be a prime ideal above the rational prime $l$, and let $\mathfrak l'$ denote its conjugate under the nontrivial automorphism of $F$. For the twisted Borcherds product value $\tilde\Psi_m(\mathcal{C}\mathcal{M}(K))$, let $p$ and $q$ be the parameters occurring in its definition. Prime-divisor conjecture. If $m$ is a positive integer and

$$\operatorname{ord}_\mathfrak l\bigl(\tilde\Psi_m(\mathcal{C}\mathcal{M}(K))\bigr)\neq 0,$$

then $\mathfrak l'\neq\mathfrak l$ and

$$4l\mid m^2p^2q-r^2$$

for some $r\in\mathbb{Z}$ satisfying

$$|r|<mp\sqrt{q}.$$

The conjecture gives an arithmetic restriction on the prime ideals dividing CM values of twisted Borcherds products, relating their valuations to the parameters of the product and to a bounded integer $r$.