Non-tautological cycles on moduli spaces of smooth pointed curves

In recent work by Arena, Canning, Clader, Haburcak, Li, Mok, and Tamborini it was proven that for infinitely many values of $g$ and $n$, there exist non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth genus $g$, $n$-pointed curves. Here we show how a generalization of their technique allows to cover most of the remaining cases, proving the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ for all but finitely many values of $g$ and $n$.