The word-map nonconstancy conjecture for simple compact Lie groups

Let $G$ be a connected simple compact Lie group with trivial center. A non-constant word is an element of $G\ast F_d$ that does not lie in $G$ itself; its evaluation on $G^d$ is the associated word map.

Word-map nonconstancy conjecture. Every non-constant word defines a non-constant word map.

The trivial-center hypothesis is necessary, since a nontrivial central element produces a non-constant word with constant evaluation. The claim also fails for certain semisimple, non-simple compact groups, while the conjecture asks for the corresponding nonconstancy property in the simple case.