On A Tautological Relation Conjectured By Buryak-Shadrin

Buryak and Shadrin conjectured a tautological relation on moduli spaces of curves M̄g,n which has the form Bᵐg, d=0 for certain tautological classes Bᵐg, d where m ≥ 2, n ≥ 1 and |d| ≥ 2g+m-1. In this paper we prove that this conjecture holds if it is true for the $m=2$ and |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.