Yes, this is precisely the content of Theorem 7.11 in [arXiv:1410.5675](https://arxiv.org/abs/1410.5675), which should be combined with §7.4 of [arXiv:1510.04969](https://arxiv.org/abs/1510.04969).
In fact, the cited results prove this for any nonsymmetric operad in chain complexes over a commutative ring,
and are also applicable to ∞-categories other than chain complexes.

In the case of symmetric operads (such as E_∞) the result continues to hold
for characteristic 0, i.e., rational chain complexes.
For symmetric operads in characteristic p there are algebras that cannot be rectified as stated,
and one must use other categories instead of chain complexes,
e.g., simplicial modules over F_p (see §7.3 of [arXiv:1510.04969](https://arxiv.org/abs/1510.04969)).