On a tautological relation conjectured by Buryak and Shadrin

Buryak and Shadrin conjectured a tautological relation on moduli spaces of curves $\overline{\mathcal{M}}_{g,n}$ which has the form $B^m_{g,~\textbf{\emph{d}}}=0$ for certain tautological classes $B^m_{g,~\textbf{\emph{d}}}$, where $m~\geq~2$, $n~\geq~1$ and $|\textbf{\emph{d}}|~\geq~2g+m-1$. In this paper, we prove that this conjecture holds if it is true for the $m=2$ and $|\textbf{\emph{d}}|~=~2g+1$ case. This result reduces the proof of this conjecture to checking finitely many cases for each genus $g$. We also prove this conjecture for the $g=1$ case.