Arity Approximation of $\mathrm{\infty }$-Operads

Let ${\mathbb{E}}_d$ denote the little discs operad for $1\le d\le \mathrm{\infty }$ and let $\mathcal{C}$ be an $\mathrm{\infty }$-category all of whose mapping spaces are $n$-truncated. We prove that when considering ${\mathbb{E}}_d$-monoids in $\mathcal{C}$, all coherence diagrams of arity $>n+3$ are redundant. More generally, for an $\mathrm{\infty }$-operad $\mathcal{O}$ we bound the arity of the relevant coherence diagrams in terms of the connectivity of certain operadic partition complexes associated to $\mathcal{O}$.