Let $R$ be a commutative ring with characteristic $0$, namely it contains the field of rational numbers. Higher Algebra Proposition 7.1.4.10 tells that the category of commutative $R$-dg-algebras $\mathrm{CAlg^{dg}}(R)$ has a model structure induced from the projective model structure on chain complexes $\mathrm{Ch}(R)$, where weak equivalences are quasi-isomorphisms and fibrations are surjections (so, a morphism of commutative dg-algebras is a fibration or a weak equivalence if its underlying morphism of chain complexes is such).

In the proof of the following Proposition 7.1.4.11, an unproved claim (a condition in 4.5.4.7) is implicitly used, namely:

The forgetful functor $\mathrm{CAlg^{dg}}(R) \to \mathrm{Ch}(R)$ preserves fibrant-cofibrant objects.

Now, every object is fibrant with respect to the considered model structures, so this claim boils down to check that if $A$ is a cofibrant object in $\mathrm{CAlg^{dg}}(R)$, then its underlying chain complex is cofibrant with respect to the projective model structure on $\mathrm{Ch}(R)$.

How can I prove this? If $R$ were a field then it would be very easy, because every chain complex over a field is cofibrant. I feel that I should somehow use that $R$ has characteristic $0$ (it contains $\mathbb Q$), but can't precisely figure out how.