Semiprojectivity of the moduli of principal $G$-bundles with $\lambda$-connections

Let $X$ be a compact connected Riemann surface of genus $g \geq 2$ and $G$ a nontrivial connected reductive affine algebraic group over $\mathbb{C}$. We prove the semiprojectivity of the moduli spaces of semistable $G$-Higgs bundles and $G$-bundles with $\lambda$-connections of fixed topological type $d\in \pi_1(G)$.
Comment: In the earlier version, Proposition 2 was incorrect, which was crucial for the main result of that version