The conjectural cohomology-ring presentation for stable maps of degree three

Let $n$ be finite, and let $A$ be the $5\times5$ matrix and $G_1$ the column vector defined above. Write $G_{n+1}=A^nG_1$. The generators $b,\sigma_1,\rho,\sigma_2,\tau,\sigma_3$ are classes in the cohomology ring of the stable-map space.

Cohomology-ring presentation conjecture. The de Rham cohomology ring is

$$H_{DR}\big(\overline{M}_{0,0}({\mathbb P}^n,3)\big)=\frac{{\mathbb C}[b,\sigma_1,\rho,\sigma_2,\tau,\sigma_3]}{\big((\tau^2-\rho\sigma_2),\tau\sigma_3,\rho\sigma_3,G_{n+1}\big)}.$$

The five moving relations in $G_{n+1}$ have degrees $n$, $n+1$, $n+1$, $n+2$, and $n+2$.

This conjecture proposes a complete presentation of the cohomology ring for finite $n$, extending the known presentation in degrees less than $n$.