The vanishing conjecture for the classes $B^m_{g,\overline{d}}$

For $m\ge 0$, $g\ge 0$, and $n\ge 1$, let $\overline{d}=(d_1,\ldots,d_n)\in\mathbb Z_{\ge 0}^n$, and let $B^m_{g,\overline{d}}$ be the tautological class defined from admissible balanced trees. It lies in $R^{\sum d_i}({\overline{\mathcal{M}}}_{g,n+m})$.

Vanishing conjecture for $B^m_{g,\overline{d}}$. For any $m\ge 2$, $g\ge 0$, $n\ge 1$, and $\overline{d}$ such that $\sum d_i\ge 2g+m-1$, one has

$$B^m_{g,\overline{d}}=0\quad\text{in }R^{\sum d_i}({\overline{\mathcal{M}}}_{g,n+m}).$$

This is the paper's first conjecture and forms part of a proposed system of tautological relations on moduli spaces of stable curves. The supplied text gives no resolution status.