On the pathwise uniqueness of stochastic 2D Euler equations with Kraichnan noise and Lᵖ-data

In the recent work [arXiv:2308.03216], Coghi and Maurelli proved pathwise uniqueness of solutions to the vorticity form of stochastic 2D Euler equation, with Kraichnan transport noise and initial data in L¹∩ Lᵖ for $p>3/2$. The aim of this note is to remove the constraint on p, showing that pathwise uniqueness holds for all L¹∩ Lᵖ initial data with arbitrary $p>1$.