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Given the prominent role of $E_{\infty}$-operads in algebraic topology it seems right to add this tag to the question (and, in particular, the paper quoted in the question is an algebraic topology paper).
I added the tags algebraic topology and homotopy theory, since this question turns out to be also a homotopy theoretic question (existence of model structure) and a question related to algebraic topology (the prominent role of simplicial rings and cdgas in this field).
Concerning the growth of Betti numbers of free loop spaces, it is worth mentionning also the papers of Pascal Lambrechts On the Betti numbers of the free loop space of a coformal space (JPAA 2001) and The Betti numbers of the free loop space of a connected sum ( J. London Math. Soc. 2001).
