Regularity of minimizing p-harmonic maps – i.e., minimizers of the Dirichlet p-energy among maps between two given manifolds – is known to depend on the topology of the target manifold. In particular, the case of maps into spheres has been studied intensively, but still some of the most basic questions concerning maps from B3 into S3 remain open. Minimizing maps in this context were shown to be regular when p=2 or p≥3, and recently also when 2<p<2.13, leaving a peculiar gap in between. I will discuss known approaches to the problem and how to prove regularity for 2<p<2.36 and 2.97<p<3, thus shrinking the gap. This is joint work with Michał Miśkiewicz (University of Warsaw).