Integral conjecture for finite-dimensional Nichols–Woronowicz algebras

Let $(W,S)$ be a finite Coxeter system, let $\EuScript{B}_W$ be its Nichols–Woronowicz algebra, and suppose that $\EuScript{B}_W$ is finite dimensional. Let $D$ be a complete disjoint system of order $r$, let $w_1,\ldots,w_r$ be an ordering of the elements of $D$, and define

$$y_i=w_ix_{w_o}\qquad (1\leq i\leq r).$$

Integral conjecture. The element

$$y_1\cdots y_r$$

is a nonzero integral in $\EuScript{B}_W$. The claim is part of a broader structural analysis of integrals in finite-dimensional Nichols–Woronowicz algebras and is presented as a conjectural consequence relevant to the paper's motivating arguments; the supplied context does not establish its status beyond the conjectural formulation.