The theta block characterization conjecture for Jacobi forms

Let $L$ be the lattice indexing the Jacobi-form space $J_{k,L,1}$, and let $\phi\in J_{k,L,1}$. Let $T_{-}(2)$ denote the index-raising Hecke operator, and let $\operatorname{Grit}(\phi)$ and $\operatorname{Borch}$ denote the Gritsenko lift and Borcherds product, respectively. Theta block characterization conjecture. The function $\operatorname{Grit}(\phi)$ has a Borcherds product expansion,

$$\operatorname{Grit}(\phi)=\operatorname{Borch}\left(-\frac{\phi\lvert T_{-}(2)}{\phi}\right),$$

if and only if $\phi$ is a pure theta block of the specified type, with $f(0,\ell)\geq 0$ for all $\ell$, and $\phi$ has vanishing order one in $q$. This is presented as a generalization of the theta block conjecture; the preceding discussion notes that particular families have been proved, but does not establish the full if-and-only-if statement.