A-class conjecture for no frozen legs

Let $Q=\mathbb{Q}[a_1,\dots,a_n]$, let $\pi\colon {\overline{\mathcal{M}}}_{g,n+1}\to {\overline{\mathcal{M}}}_{g,n}$ forget the last marked point, and define

$$A^0_{g,n}=\frac{\pi_*A^1_{g,n}}{a_1+\cdots+a_n}.$$

Let ${}^{lvl}\Omega^0_{g,n}$ be the degree-labeled stable tree expression with no frozen legs. A-class conjecture. For $g\geq 0$ and $n\geq 1$, one has

$$\deg\bigl({}^{lvl}\Omega^0_{g,n}-A^0_{g,n}\bigr)\leq 2g-2.$$

The source notes that $A^0_{g,n}$ is polynomial in the $a_i$ and presents this as the A-class conjecture for no frozen legs. The supplied text gives no resolution of the general claim.