We determine all proper holomorphic maps of balls $${\mathbb {B}}_2 \rightarrow {\mathbb {B}}_3$$ B 2 → B 3 admitting a $$C^3$$ C 3 extension up to the boundary of $${\mathbb {B}}_2$$ B 2 and whose boundary values $$S^3 \rightarrow S^5$$ S 3 → S 5 are subelliptic harmonic maps (in the sense of Jost and Xu in Trans Am Math Soc 350(11):4633–4649, 1998). A new numerical CR invariant, the CR degree of a CR map of spheres $$S^{2n+1} \rightarrow S^{2N+1}$$ S 2 n + 1 → S 2 N + 1 , is introduced and used to distinguish among the spherical equivalence classes in Faran’s list $$P^*(2,3)$$ P ∗ ( 2 , 3 ) (cf. Faran in Invent Math 68:441–475, 1982). As an application, the boundary values $$\phi $$ ϕ of Alexander’s map $$\Phi \in P(2,3)$$ Φ ∈ P ( 2 , 3 ) (cf. Alexander in Indiana Univ Math J 26:137–146, 1977) is shown to be homotopically nontrivial, as a map of $$\{ (z,w) \in S^3 : w + \overline{w} > 0 \}$$ { ( z , w ) ∈ S 3 : w + w ¯ > 0 } into $$S^5 {\setminus } S^3$$ S 5 \ S 3 .