Hodge-proper stack conjecture on rational étale cohomology

Let $K$ be a finite extension of $\mathbb Q_p$, let $\mathcal O_K$ be its ring of integers, and let $\mathpzc X$ be a smooth Hodge-proper stack over $\mathcal O_K$. Let $\widehat{\mathpzc X}$ denote its $p$-adic completion, and let $\Upsilon_{\mathpzc X}$ be the natural comparison map. Hodge-proper stack conjecture.

$$\Upsilon_{\mathpzc X}\colon R\Gamma_{\mathrm{\acute et}}(\mathpzc X_{\mathbb C_p},\mathbb Q_p)\longrightarrow R\Gamma_{\mathrm{\acute et}}(\widehat{\mathpzc X}_{\mathbb C_p},\mathbb Q_p)$$

is an equivalence. This would identify the rational étale cohomology of a Hodge-proper stack with that of its completion. Equality of the dimensions of the individual cohomology groups provides evidence, but the equivalence remains conjectural.