Conformal harmonic maps in high dimensions are scaled isometries

Question

It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.

I discovered lately that in dimension $d>2$, conformal harmonic maps must be scaled isometries (The conformal factor is constant).

This is a "rigidity phenomena"- in stark contrast to the $2$D case, in higher dimensions the set of conformal and harmonic maps is very "small".

I am quite sure this should be already known, but couldn't find a reference. Any help?

Answer 1

After posting in MO, it turns out that:

Indeed, the result can be found in the book Harmonic morphisms between Riemannian manifolds, by Paul Baird, John C. Wood.

The relevant statement is Corollary 3.5.2.