Theta blocks related to root systems

Gritsenko, Skoruppa and Zagier associated to a root system $R$ theta block $\vartheta_R$, which is Jacobi form of lattice index. We classify the blocks $\vartheta_R$ $q$-order $1$ show that their Gritsenko lift strongly-reflective Borcherds product singular weight, related Conway's group $\operatorname{Co}_0$. As corollary we obtain proof conjecture by Poor Yuen for pure obtained as specializations functions $\vartheta_R$.