Contractibility of the space of generic opers for classical groups

Let $G$ be a reductive group and $X$ a smooth projective curve. We prove that, for $G$ classical and $\sigma$ an arbitrary $G$-local system on $X$, the space $\overline{\operatorname{Op}}^{gen}_{G,\sigma}$ of generic extended oper structures on $\sigma$ is homologically contractible. This contractibility result is crucial for the proof of the geometric Langlands conjecture.