Theta blocks related to root systems

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