Genus-$g$ spinor Eisenstein-series conjecture for lattice integrals

Let $I_d^g$ be the integral of the $(d,d)$ lattice partition function over the fundamental domain of the moduli space of genus-$g$ Riemann surfaces, and let $\mathcal{E}^{SO(d,d,\mathbb{Z})}_{\mathbf S;s=g}$ and $\mathcal{E}^{SO(d,d,\mathbb{Z})}_{\mathbf C;s=g}$ be the order-$g$ Eisenstein series in the spinor and conjugate-spinor representations. Genus-$g$ threshold conjecture. Up to an overall factor,

$$I_d^g\propto\mathcal{E}^{SO(d,d,\mathbb{Z})}_{\mathbf S;s=g}+\mathcal{E}^{SO(d,d,\mathbb{Z})}_{\mathbf C;s=g}.$$

The conjecture follows from a Laplace-eigenvalue comparison under an explicitly stated plausible integration-by-parts assumption; the source gives no resolution evidence.