In the "ordinary" operad category, it is known that there is a colored operad $Op$ with set of colors $\mathbb{N}$ corresponding to "degrees" of vertices and with operations indexed by trees, such that algebras over $Op$ in $\mathrm{Set}$ (or more generally any symmetric monoidal category) correspond to monochromatic operads.
Is it known that $\infty$-algebras over $Op$ are equivalent to monochromatic $\infty$-operads?
Yes, combine Corollary 9.4.1 and Theorem 7.11 of arXiv:1410.5675, for example.
This topic is also examined more explicitly in the work of Chu and Haugseng, arXiv:1707.08049.
Corollary 5.1.13 shows that enriched ∞-operads are equivalent to ∞-algebras over the operad of colored operads.
Theorem 5.2.10 proves (using arXiv:1410.5675) that the underlying ∞-category of the relative category of ordinary enriched colored operads is equivalent to the ∞-category of enriched colored ∞-operads.
