Singular-weight Borcherds products conjecture

Let $F$ be a Borcherds product of singular weight for an even lattice $M$ of signature $(2,n)$ with $n\geq 3$. Singular-weight Borcherds products conjecture. There exists an even lattice $M'$ such that

$$M'\otimes \mathbb Q=M\otimes \mathbb Q$$

and $F$ can be viewed as a reflective modular form for $M'$. The conjecture is motivated by the fact that all known singular-weight Borcherds products are reflective apart from certain pull-backs, including an example that is not reflective for the original lattice; its proposed conclusion allows replacing the lattice by another even lattice in the same rational quadratic space.