Pro-étale Poincaré lemma for topological derived de Rham complexes

Let $X$ be a smooth pseudo-rigid analytic space over $\mathcal{O}_K[[t]]\left<\pi^a/t^b\right>[1/t]$. Consider the projective maps

$$g:X_{\text{pro-étale}}\rightarrow X_{\text{ét}},\qquad f:X_{\text{pro-étale}}\rightarrow X.$$

Pro-étale Poincaré conjecture. The construction in this section should extend to the pro-étale sites, giving two strictly exact long exact sequences of topological derived de Rham complexes, one relative to $X$ and one relative to $X_{\text{ét}}$, as displayed in the source.

This is presented as an analogue of the rigid-analytic result cited in the paper. The notation for the displayed complexes and the precise meaning of the asserted exactness are not fully defined in the supplied span.