I know of three homotopy theories of colored operads.

- The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak equivalences given by strict maps $O\to O'$ which induce weak equivalences on spaces of operations.
- The "dendroidal" notion of infinity-operad, introduced by Moerdijk and Weiss and further studied in this paper by Cisinski and Moerdijk.
- Lurie's infinity-operads, which are infinity categories fibered over the nerve of the category $\mathrm{Set}^*$ of pointed sets, satisfying certain Segal-style properties.

If all is well in the world, these three homotopy theories (viewed as e.g. $\infty$-categories) should be equivalent (or there should be a good reason for them not to be). But I can't find references for any equivalences between them. This paper seems to compare (1) and (2), via a Quillent adjunction, which it does not show is an equivalence. The obvious functor (1) $\implies$ (3) is written down in Lurie's Higher Algebra, but it is not (as far as I can tell) shown to be an equivalence.

Is more known about comparisons between these homotopy theories? I'm specifically interested in the comparison between (1) and (3). Are the corresponding $\infty$-categories equivalent, perhaps under additional restrictions? Is the functor (1) $\implies$ (3) fully faithful? Are there known examples where this functor is not an equivalence?