Folklore Koszul self-duality conjecture for operads and right modules

Let $O$ be a termwise $\Sigma$-finite, termwise cofibrant operad and let $R$ be a termwise $\Sigma$-finite, termwise cofibrant right module in $(\mathrm{Sp},\wedge)$. Write $K$ for Koszul duality. Folklore conjecture. There is a natural zigzag of equivalences of operads, together with a compatible natural zigzag of equivalences of modules,

$$O \simeq \dots \simeq K(K(O))$$ $$R \simeq \dots \simeq K(K(R)).$$

This conjecture proposes involutivity of Koszul duality for suitably finite and cofibrant operads and right modules, and is used in the paper to motivate the corresponding collapse-map construction in stable, framed embedding calculus.