The theta block conjecture for Borcherds products

Let the pure theta block $\Theta_f$ be a holomorphic Jacobi form of weight $k$ and index $m$ with vanishing order $1$ in $q$, where $k,m\in\mathbb Z_{>0}$. Define the nearly holomorphic Jacobi form

$$\Psi_f=-\frac{\Theta_f|T_-(2)}{\Theta_f}$$

of weight $0$ and index $m$, where $T_-(2)$ is the index-raising Hecke operator. Theta block conjecture. Every such pure theta block satisfies

$$G(\Theta_f)=B(\Psi_f).$$

Here $G$ denotes the Gritsenko lift and $B$ the multiplicative Borcherds lift. The conjecture gives a sufficient condition for a Gritsenko lift to be a Borcherds product and was formulated by Gritsenko, Poor and Yuen; its resolution is not established by the supplied text.