  In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$
to $\mathbb{S}^2$. We show that every boundary map can be modified to a
boundary map that admits multiple minimizers of the Dirichlet energy by a small
$W^{1,p}$-change for $p<2$. This strengthens a remark by the second-named
author and Strzelecki. The main novel ingredient is a homotopy construction,
which is the answer to an easier variant of a challenging question regarding
the existence of a norm control for homotopies between $ W^{1,p} $ maps.
