In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian fibration of simplicial sets $C^\otimes\to N(\Delta)^{op}$, whereas the latter replaces $N(\Delta)^{op}$ with $\mathcal{A}ss^\otimes$, Lurie's associative $\infty$-operad. The former is perhaps more easily used because of Proposition 1.6.3 of DAG II which allows us to produce a cocartesian fibration of simplicial sets $N(M^\otimes)\to N(\Delta^{op})$ whenever we started with a topologically enriched monoidal category $M$.

Unfortunately, this does not immediately provide us with a cocartesian fibration $N(M^\otimes)\to \mathcal{A}ss^\otimes$, where $\mathcal{A}ss^\otimes$ is the associative $\infty$-operad of Higher Algebra. There is certainly a morphism of simplicial sets $N(M^\otimes)\to \mathcal{A}ss^\otimes$, but it's not clear to me that it's necessarily easy to check whether or not this actually gives the structure I desire (that is, a monoidal structure on the $\infty$-category $N(M)$ corresponding to the monoidal structure on $M$). It happens to follow from Theorem 4.7.1.10 of Higher Algebra that if the map of marked simplicial sets $N(M^\otimes)^\natural\to (N(\Delta)^{op})^\natural$ (where I've used $\natural$ to indicate marking the inert morphisms in $N(\Delta)^{op}$ according to Definition 4.7.1.9 of HA and the inert morphisms of $N(M^\otimes)$ simply being the ones that project onto inert morphisms of $N(\Delta)^{op}$) is *fibrant* in the model structure on $(\mathcal{P}Op_\infty)_{/(N(\Delta)^{op})^\natural}$ induced from the model structure on $\mathcal{P}Op_\infty$, then the above composition produces a planar $\infty$-operad. However, it seems like the only way to check if something is fibrant is to know beforehand that it actually comes from a planar $\infty$-operad.

So my question is the following: given a topologically enriched monoidal category $M$, is there a planar $\infty$-operad $p:N(M^\otimes)\to\mathcal{A}ss^\otimes$ such that the fiber over $\langle 1\rangle$ is equivalent to $N(M)$ and such that a strict algebra in $M$ gives an associative algebra of $p:N(M^\otimes)\to\mathcal{A}ss^\otimes$?

Thanks! :-)