Exact U-duality Eisenstein-series conjecture for toroidal type II $R^4$ couplings

Let type II string theory be compactified on $T^d$, equivalently M-theory on $T^{d+1}$, and let $E_{d+1(d+1)}(\mathbb{Z})$ be the U-duality group. Denote by $\mathcal{E}^{E_{d+1(d+1)}(\mathbb{Z})}_{\mathrm{string};s=3/2}$ the Eisenstein series in the string-multiplet representation. Exact $R^4$ coupling conjecture. The exact four-graviton $R^4$ coupling is, up to a factor of Newton's constant,

$$f_{R^4}=\frac{V_{d+1}}{l_M^9}\mathcal{E}^{E_{d+1(d+1)}(\mathbb{Z})}_{\mathrm{string};s=3/2}.$$

This is proposed as a non-perturbative U-duality completion; the source supplies no resolution evidence.