In some different places (e.g. in Berger & Fresse, Combinatorial operad actions on cochains), it is stated as a classical result that the Barratt-Eccles operad $\mathcal{E}$ (in chain complexes) is $\Sigma_* $-cofibrant (i.e. not cofibrant as an operad, but as symmetric sequence) and that the (arity-wise) tensor product $\mathcal{E} \otimes \mathcal{O}$ is $\Sigma_*$-cofibrant for any chain operad $\mathcal{O}$.
Can someone point me in the direction of why this is true? Does it already follow from the paper of Barratt & Eccles? Or can the respective lifts just be constructed by hand?