Buryak and Shadrin conjectured a tautological relation on moduli spaces of curves \(\overline{\mathcal{M}}_{g,n}\) which has the form B g,d m = 0 for certain tautological classes B g,d m , 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 also prove this conjecture for the g = 1 case.
