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Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of $\mathcal{S}$-filtered objects.

If $\mathcal{G}$ is a Grothendieck category, it is known that $\mathcal{G}$ is deconstructible in itself.

Question: if $\mathcal{G}$ is a Grothendieck category, let $\mathcal{E}$ be the class of all acyclic complexes in $\mathrm{Ch}(\mathcal{G})$. Is it true that $\mathcal{E}$ is deconstructible in $\mathrm{Ch}(\mathcal{G})$?

I found some results in which this fact was used more or less explicitly, so I believe it is true. However, I couldn't find any proof. Does anybody know of a (elementary) proof? Is it possible to exhibit a set $\mathcal{S}$ such that $\mathrm{Filt}\,\mathcal{S} = \mathcal{E}$?

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This is proved in Proposition 4.4 of "Deconstructibility and the Hill Lemma in Grothendieck categories" by Jan Šťovíček.

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