Tautological generation conjecture for Higgs-bundle stacks on smooth curves

Let $C$ be a smooth complex curve, and let $\cT_n(C)$ be the stack considered in the source, with projection $\pi$ and universal bundle $\cE$. Define the subring

$$TH^*(\cT_n(C),\mathbb Z)\subset H^*(\cT_n(C),\mathbb Z)$$

generated by the classes $\pi_*(c_i(\cE)\cup\gamma)$ for $i\geq 0$ and $\gamma\in H^*(C,\mathbb Z)$. Let $\Lambda$ be the coefficient ring acting on this cohomology, and let $\cF_n(C)$ denote the full-flag space with its pushforward map to $\cT_n(C)$.

Tautological generation conjecture. The following assertions hold:

1. $H^*(\cT_n(C),\mathbb Z)$ is generated by $TH^*(\cT_n(C),\mathbb Z)$ over $\Lambda$.
2. The image of

$$H^*(\cF_n(C),\mathbb Z)\longrightarrow H^*(\cT_n(C),\mathbb Z)$$

is precisely $TH^*(\cT_n(C),\mathbb Z)$.

For $C=\mathbb P^1$, the source proves the analogous statement using tautological classes and the full-flag space. The conjecture proposes the corresponding generation and image description for an arbitrary smooth complex curve.