Critical-weight and lattice-index theta-block conjecture for intermediate Lie algebras

Let $\mathfrak{g}_I$ be an intermediate Lie algebra between $\mathfrak{g}_1$ and $\mathfrak{g}_2$, and let the associated theta block be constructed from the positive roots and weights of $\mathfrak{g}_I$. Its critical weight is $\big(\operatorname{rk}(\mathfrak{g}_1)+1\big)/2$, and its lattice index is the index determined by the normalized bilinear form in the construction. Critical-weight and lattice-index theta-block conjecture. The theta blocks of the specified type associated with the intermediate Lie algebras in the intermediate exceptional series are holomorphic Jacobi forms of critical weight and lattice index. This refines the earlier critical-weight proposal by specifying the lattice index; it remains unproved in the supplied text.