Buryak–Shadrin tautological vanishing conjecture

Let $g,m\geq 0$, $n\geq 1$, and let $\textbf{d}=(d_1,\ldots,d_n)\in(\Bbb Z_{\geq 0})^n$, with $|\textbf{d}|:=d_1+\cdots+d_n$. For a nondegenerate balanced genus-$g$ stable rooted tree $T$ with $n$ regular legs, $m$ frozen legs, and possibly extra legs, let

$$B^m_{g,\textbf{d}}:=\sum_{T\in{\operatorname{SRT}}^{(b,nd)}_{g,n,m;o}}(-1)^{|E(T)|}e_*[T,q_{\textbf{d}}]\in H^{2|\textbf{d}|}(\overline{\mathcal{M}}_{g,n+m}),$$

where $e$ forgets the marked points corresponding to extra legs and $|E(T)|$ is the number of edges of $T$. Buryak–Shadrin conjecture. For any $g\geq 0$, $m\geq 2$, $n\geq 1$, and $n$-tuple of nonnegative integers $\textbf{d}$ such that $|\textbf{d}|\geq 2g+m-1$, one has

$$B^m_{g,\textbf{d}}=0.$$

This is a tautological relation on the moduli space of stable curves. The source notes that Buryak and Shadrin proved the relation when either $n=1$ or $g=0$, while the general statement is presented as their conjecture.