On A Tautological Relation Conjectured By Buryak-Shadrin

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