The transcendence conjecture for the KZ associator

Let $M$ be the scheme of associators and let $(2\pi\operatorname{i},\Phi_{KZ})\in M(\mathbb C)$ be the KZ associator. Its coordinate-ring point defines a morphism

$$\varphi_{KZ}:{\mathbb Q}[M]\to\mathbb C.$$

Transcendence Conjecture. The morphism $\varphi_{KZ}$ is injective. This asserts algebraic independence of the coordinate functions of the associator in the sense encoded by the coordinate ring of $M$; the source provides no evidence of a resolution, so the conjecture remains open.