Conjecture on independence of essential tautological classes

Let $g$ be a nonnegative integer, let $P$ be a finite set of markings, and let $k$ be a nonnegative integer. Consider the essential tautological classes in $H^{2k}\left(\overline{\mathcal{M}}_{g,P}\right)$, meaning tautological classes not eliminable by relations among tautological classes arising in lower degree. Independence conjecture. There are no relations between essential tautological classes in $H^{2k}\left(\overline{\mathcal{M}}_{g,P}\right)$ whenever $g\geq 3k$. The conjecture is motivated by the proof of the preceding dimension result and is compatible with Harer's and Ivanov's stability theorems and with Faber's results and conjectures for the tautological ring of $\mathcal{M}_{g,n}$.