The nonexistence conjecture for proper complex-valued $(p,1)$-harmonic morphisms

Let $p\ge 2$ and let $\phi:U\to\mathbb{C}$ be a complex-valued $(p,1)$-harmonic morphism defined locally on the standard Euclidean space $\mathbb{R}^{2p-1}$. The nonexistence conjecture. Then $\phi$ is a $(1,1)$-harmonic morphism, that is, $\tau(\phi)=0$. This would show that no proper $(p,1)$-harmonic morphism exists locally from $\mathbb{R}^{2p-1}$ to $\mathbb{C}$; the source gives no proof or resolution of the claim.