Degeneracy conjecture for string, particle and membrane U-duality Eisenstein series

Let $d>2$, let $E_{d+1(d+1)}(\mathbb{Z})$ be the U-duality group, and let $\mathcal{E}_{\mathrm{string};3/2}$, $\mathcal{E}_{\mathrm{particle};d/2-1}$, and $\mathcal{E}_{\mathrm{membrane};1}$ denote its Eisenstein series in the indicated multiplet representations and orders. String–particle–membrane degeneracy conjecture. These series are equal up to the displayed normalization:

$$\frac{V_{d+1}}{l_M^9}\mathcal{E}_{\mathrm{string};3/2}=\frac{\Gamma(d/2-1)}{\pi^{d/2-2}}\mathcal{E}_{\mathrm{particle};d/2-1}=\frac{V_{d+1}}{l_M^9}\mathcal{E}_{\mathrm{membrane};1}.$$

The three series have the same Laplace eigenvalue, motivating the conjecture, but the source gives no proof of equality.