Proof.
The Segal conditions for \(\infty\)-operads imply that surjectivity on underlying functors is equivalent to surjectivity on \(\infty\)-categories of colors. Moreover, the hom-spaces in an \(\infty\)-operad are disjoint unions of (finite) products of multi-hom spaces, and these operations both preserve the class of \((n-1)\)-connected (resp. \((n-1)\)-truncated) morphisms of spaces. (To see that products preserve \((n-1)\)-connectedness (resp. \((n-1)\)-truncatedness), note that this notion is determined fiberwise, that fibers of a product of morphisms of spaces are computed factorwise since limits commute with limits, and that \((n-1)\)-truncated (resp. \((n-1)\)-connected) spaces are stable under products.) ◻