The 1/4-BPS black-hole index formula from the next-to-minimal representation of E8

Let $\varphi_{\mathrm{ntm}}$ be an automorphic form in the next-to-minimal representation $\pi_{\mathrm{ntm}}$ of $E_8$, and let $(p,q)$ denote electric-magnetic charges parametrising the character $\psi_p$. Let $\Omega_{1/4}(p,q)$ denote the index counting charged 1/4-BPS black holes.

1/4-BPS index conjecture. The index is

$$\Omega_{1/4}(p,q)=\prod_{p<\infty}F_{\psi_p}(\varphi_{\mathrm{ntm}},1).$$

This is the proposed next-to-minimal analogue of the 1/2-BPS formula, relating the black-hole index to local Fourier coefficients. The source gives no resolution of the assertion.