Invariant bilinear form conjecture for finite-dimensional irreducible $\boxtimes_q$-modules

Let $V$ be a finite-dimensional irreducible $\boxtimes_q$-module of type $1$, and let $\omega$ be the antiautomorphism of $\boxtimes_q$ from Lemma. Invariant-form conjecture. There exists a nondegenerate symmetric bilinear form $(\,,\,)$ on $V$ such that

$$(r.u,v)=(u,\omega(r).v)\qquad r\in\boxtimes_q,\quad u,v\in V.$$

Such a form would make every finite-dimensional irreducible module of type $1$ self-dual with respect to the specified antiautomorphism. The supplied text gives no resolution status.