On A Tautological Relation Conjectured By Buryak-Shadrin

Buryak and Shadrin conjectured a tautological relation on moduli spaces of curves ${\bar{\mathcal{M}}}_{g,n}$ which has the form $B_{g,\mathbf{\text{d}}}^m=0$ for certain tautological classes $B_{g,\mathbf{\text{d}}}^m$ where $m\ge 2,n\ge 1$ and $\mathrm{\mid }\mathbf{\text{d}}\mathrm{\mid }\ge 2g+m-1$. In this paper we prove that this conjecture holds if it is true for the $m=2$ and $\mathrm{\mid }\mathbf{\text{d}}\mathrm{\mid }=2g+1$ 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.