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 $L^1\cap L^p$ for $p>3\mathrm{/}2$. The aim of this note is to remove the constraint on $p$, showing that pathwise uniqueness holds for all $L^1\cap L^p$ initial data with arbitrary $p>1$.