The complete-relations conjecture for the universal subring of Higgs-bundle cohomology

Let $\mathcal{M}$ be the moduli space above, let $H_I^*(\mathcal{M})\subset H^*(\mathcal{M})^\Gamma$ be the subring generated by the universal classes $\alpha_{\mathcal{M}}$, $\beta_{\mathcal{M}}$, and $\gamma_{\mathcal{M}}$. Define

$$\rho_{r,s,t}=\sum_{i=0}^{\min(r,s)}\binom{r}{i}\binom{g-t-i}{g-t-s}\alpha^{r-i}\beta^{s-i}(2\gamma)^{t+i}$$

for $r,s,t\geq 0$, and let $R$ be the graded ring generated by $\alpha$, $\beta$, and $\gamma$ of degrees $2$, $4$, and $6$, respectively, with relations $\rho_{r,s,t}$ whenever $r+3s+3t>3g-3$. The complete-relations conjecture. The subring $H_I^*(\mathcal{M})$ is isomorphic to $R$.

The conjecture would give a complete description of the subring generated by $\alpha_{\mathcal{M}}$, $\beta_{\mathcal{M}}$, and $\gamma_{\mathcal{M}}$. It is based on numerical computer calculations; the source does not state a proof or a resolution.