Regularity of minimizing $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$

We prove that every minimizing $p$-harmonic map from $B^3$ into $\mathbb S^3$ is locally $C^{1,α}$ for some $α\in(0,1)$, for every $p>p_0$, where $p_0=\frac{7-\sqrt{17}}{2}\approx 1.44$. This closes the gap between the previously established regularity ranges $[2,2.642]\cup[2.961,3]$ and, in particular, yields full interior regularity for all $p\ge2$. The result also extends regularity to the subquadratic range $p_0<p<2$.