Let $C$ be a combinatorial symmetric monoidal model category and let $A$ be a associative algebra object in $C$, that is cofibrant as an object in $C$. In Higher Algebra 4.3.3.17, Lurie proves an equivalence of infinity-categories
$$N(Mod_A(C)^c[W^{-1}]) \cong Mod_A(N(C^c)[W^{-1}]$$
(at the left we have the $\infty$-category associated to $A$-modules in $C$; at the right we have $A$-modules (in the sense of $\infty$-categories) in the $\infty$-category associated to $C$).
If we additionally assume $A$ to be commutative, and $\otimes$ in $C$ to preserve colimits, both sides admit a symmetric monoidal structure (at the left given by the tensor product $- \otimes_A -$ in the model category $C$, at the right by 4.5.2.1 in Higher Algebra).
It is natural to expect the above equivalence to be an equivalence of symmetric monoidal $\infty$-categories in this situation. Is this assertion appearing in the literature?
(I wish to avoid imposing a condition such as in §4.5.4 in HA, which is a means to ensure that commutative algebra objects in $N(C^c)[W^{-1}]$ can be rectified.)
