  The purpose of this work is to demonstrate that the lack of selection by
smooth regularisation for the continuity equation with a bounded,
divergence-free vector field as demonstrated in \cite{DeLellis_Giri22} by De
Lellis and Giri takes place over a dense set of vector fields. More precisely,
we construct a set of bounded vector fields $D$ dense in $L^p_{loc}([0,2]\times
\R^2;\R^2)$ such that for each vector field $\bb\in D$, there are two smooth
regularisations of $\bb$, for which the unique solution of the Cauchy problem
for the continuity equation along each regularisation converges to two distinct
solutions of the Cauchy problem along $\bb$.
