A-class conjecture for one frozen leg

Let $Q=\mathbb{Q}[a_1,\dots,a_n]$. For a stable rooted tree with one frozen leg, let $A^1_{g,n}\in R^*({\overline{\mathcal{M}}}_{g,n+1})\otimes_{\mathbb{Q}}Q$ be the sum of the classes $A(T)$ defined above, and let ${}^{lvl}\Omega^1_{g,n}$ be the corresponding degree-labeled stable tree expression. Here $\deg$ denotes the degree in the variables $a_1,\dots,a_n$. A-class conjecture. For $g\geq 0$ and $n\geq 1$, one has

$$\deg\bigl({}^{lvl}\Omega^1_{g,n}-A^1_{g,n}\bigr)\leq 2g-1.$$

The conjecture is supported by computations for $(g,n)=(1,2)$ and $(2,2)$; it is proved in genus $0$ and for $n=0,1$ in the ranges stated in the supplied theorems, while the general case remains open.