The G2 quaternionic-discrete-series conjecture for Donaldson–Thomas invariants

Let $X$ be a Calabi-Yau 3-fold with $h_{1,1}(X)=1$, let $\Omega(\gamma)$ denote its Donaldson–Thomas invariants, and let $G_2$ be the split real Lie group with an arithmetic subgroup $G_2(\mathbb{Z})$ serving as the expected U-duality group.

G2 DT-invariant conjecture. There exist Calabi-Yau 3-folds $X$ with $h_{1,1}(X)=1$ whose invariants $\Omega(\gamma)$ are captured by Fourier coefficients of automorphic forms attached to the quaternionic discrete series of $G_2$.

The claim is presented in connection with the analysis of quaternionic discrete series by Gan, Gross and Savin. The supplied text gives no resolution.