Let $K$ be a perfect field with characteristic $>0$ and $\mathcal{H}$ the category of graded connected abelian hopf algebras over $K$. By a theorem of Schoeller there is a canonical equivalence between $\mathcal{H}$ and the category of Dieudonne modules over $K.$

Question. Is there a homotopy-coherent version of Dieudonné modules and a homotopy-coherent version of Schoeller's theorem that identifies a derived version of Dieudonné module with a connected $E_\infty$-hopf algebra over $K$?

By definition, an $E_\infty$-hopf algebra over $K$ is an abelian group object (in the derived sense) in the $\infty$-category of $E_\infty$-coalgebras in the derived $\infty$-category of $K,$ which is connected if its zeroth homology is canonically $K.$