Vanishing conjecture for stable tree expressions with at least two frozen legs

Let $Q=\mathbb{Q}[a_1,\dots,a_n]$, and let $\Omega^m_{g,n}\in R^*({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q$ be the class obtained by summing the tree contributions defined above. Here $\deg$ denotes the degree in the variables $a_1,\dots,a_n$. Vanishing conjecture. For $g\geq 0$, $n\geq 0$, and $m\geq 2$, one has

$$\deg \Omega^m_{g,n}\leq 2g-2+m.$$

This predicts vanishing of the coefficients above the indicated polynomial degree for stable tree expressions with at least two frozen legs. The supplied text does not state whether the conjecture has been proved or disproved.