Gritsenko–Poor–Yuen theta block conjecture for paramodular forms

Let $\Theta_f$ be a theta block of $q$-order one that is a holomorphic Jacobi form of integral weight $k$ and integral index $N$ for $A_1$. Let $\operatorname{Grit}(\Theta_f)$ denote its Gritsenko lift, let $T_{-}(2)$ be the relevant Hecke operator, and let $\operatorname{Borch}$ denote the Borcherds product lift. Gritsenko–Poor–Yuen's theta block conjecture. As a Siegel paramodular form of weight $k$ and level $N$, the Gritsenko lift is a Borcherds product, more precisely

$$\operatorname{Grit}(\Theta_f)=\operatorname{Borch}\left(-\frac{\Theta_f|T_{-}(2)}{\Theta_f}\right).$$

The claim characterizes paramodular forms that are simultaneously Gritsenko lifts and Borcherds products. The supplied status evidence says that when $k\geq4$ every theta block of $q$-order one is holomorphic and belongs to the stated $8A_1$ family or a quasi-pullback, so this conjecture is proved in that range.