Majid's conjecture on the quadratic differential algebra of simply-laced Weyl groups

Let $W$ be a Weyl group, let ${\cal C}$ be its set of reflections, and let $\Lambda_w$ be the left-invariant differential algebra associated with this differential structure. Let $\Lambda^1$ be its space of left-invariant one-forms, and define the quadratic algebra

$$\Lambda_{\quad}:=T_K\Lambda^1/\operatorname{ker}(1-\Psi),$$

where $\Psi$ is the braiding. For a simply-laced root system, Majid's conjecture.

$$\Lambda_w\cong\Lambda_{\quad}.$$

For type $A$ root systems, this conjecture was stated by S. Majid. It asserts that the quadratic relations determine the full left-invariant differential algebra in the simply-laced case; its status is not specified in the source.