Non-unimodular lattice conjecture for index-one weak Jacobi forms

Let $L$ be an even positive definite lattice, meaning $(v,v)\in2\mathbb Z$ for $v\in L$, and assume that $L$ is irreducible and not unimodular. Consider the free module of weak Jacobi forms of integral weight and index one associated to $L$, over the ring of modular forms. Non-unimodular lattice conjecture. This free module is generated by forms of non-positive weight, and it has exactly one generator of weight zero. The conjecture extends the known index-one description for unimodular lattices to non-unimodular lattices; the source gives no resolution.