Let $\mathbf{sSet}$ be the category of simplicial sets. Is it possible to put a new model structure on $\mathrm{E}_{\infty}$-algebra (of simplicial sets) such that the weak equivalences and fibrations are those defined by Joyal for quasi-categories ($\infty$-categories)?
To be more clear, I'll explain the motivation. Lets start with some notations: $\mathbf{sSet}^{K}$ is the standard model structure on simplicial sets where the fibrant objects are Kan complexes and $E_{\infty}^{K}$ the standard $E_{\infty}$-operad on $\mathbf{sSet}^{K}$. Lets $\mathbf{M}$ be the category of $E_{\infty}^{K}$-algebras.
Lets $\mathbf{sSet}^{Q}$ be the category of simplicial sets with the Joyal model structure (quasi-categories).
First question: Suppose the $(\mathrm{C},\otimes)$ is symmetric monoidal enriched category over $\mathbf{sSet}$, is it true that the coherent nerve $N\mathrm{C}$ is an $E_{\infty}^{K}$-algebra?
second question: Can we put a model structure on $\mathbf{M}$ such that the weak equivalences (fibrations) are weak equivalences (fibrations) of $\mathbf{sSet}^{Q}$?
