  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 $p>3/2$.
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 $p>1$.
