On the pathwise uniqueness of stochastic 2D Euler equations with Kraichnan noise and L^p-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 L^1∩ L^p for p>3/2. The aim of this note is to remove the constraint on p, showing that pathwise uniqueness holds for all L^1∩ L^p initial data with arbitrary p>1.