Algebraic refinement conjecture for the multivariable theta function

Let $b$ be a symmetric bilinear form and let $\theta_b$ be the holomorphic section of the complex Looijenga line bundle $\mathcal{L}_b^{\mathbb{C}}\otimes\omega_{\mathbb{C}}^{\otimes d/2}$ over the complex universal elliptic curve, with $\mathcal{L}_b^{\mathbb{Z}}$ its algebraic integral refinement. Algebraic theta-function refinement conjecture. The section $\theta_b$ refines to an algebraic section of $\mathcal{L}_b^{\mathbb{Z}}\otimes\omega^{\otimes d/2}$. This would provide an integral algebraic refinement of the analytic multivariable theta function and support the proposed connection between derived Looijenga line bundles and TMF.