Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

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.