Let $\mathcal{C}$ be an $\infty$-operad. Then Lurie in Higher Algebra, section 3.3.3 constructs a family of $\infty$-operads $$\operatorname{Mod}(\mathcal{C})^\otimes\to \operatorname{Fin}_\ast \times \operatorname{CAlg}(\mathcal{C})$$ parametrizing the operad of modules over a commutative algebra.
It seems to me that there should be an alternative construction of this family: let $\mathcal{CM}^\otimes$ be the operad defined in this paper by Saul Glasman. Then it is known that the $\infty$-category of algebras $\operatorname{Alg}_{\mathcal{CM}}(\mathcal{C})$ is equivalent to the underlying $\infty$-category $\operatorname{Mod}(\mathcal{C})$ of the above family.
I am inclined to conjecture that there exists a homotopy cartesian diagram of $\infty$-categories
$$\require{AMScd} \begin{CD} \operatorname{Mod}(\mathcal{C})^\otimes @>>> \operatorname{Alg}_{\mathcal{CM}^\otimes}(\mathcal{C})^\otimes \\ @VVV @VVV \\ \operatorname{Fin}_\ast\times \operatorname{CAlg}(\mathcal{C}) @>>> \operatorname{CAlg}(\mathcal{C})^\otimes \end{CD}$$
where the bottom arrow is the approximation of operads of HA.2.4.3.6, and the operad structure on algebras is the pointwise tensor product defined in HA.3.2.4. The intuition behind this formula is that it should encode the equivalence $$A\otimes_{A\otimes A}(M\otimes N)\simeq M\otimes_A N$$ for any commutative algebra $A$ and $A$-modules $M$ and $N$.
Is the above conjecture true? If so what is a reference?
