Linear independence conjecture for the quantum unitary group $U_{q,\Theta}(n)$

Let $C(U_{q,\Theta}(n))$ be the universal $C^*$-algebra generated by $\{\mathscr{D}^{-1}\}\cup\{V_{i,j}:1\leq i,j\leq n\}$ satisfying the defining relations above, and let $\mathbb{C}[U_{q,\Theta}(n)]$ be the corresponding $*$-algebra. Write $\tilde{B}$ for the specified basis of $\mathbb{C}[U_{q,\Theta}(n)]$. A representation is a $*$-algebra homomorphism into $C(U_{q,\Theta}(n))$. Linear independence conjecture. There exists a representation

$$\Finv:\mathbb{C}[U_{q,\Theta}(n)]\longrightarrow C(U_{q,\Theta}(n))$$

such that the set $\{\Finv(b):b\in\tilde{B}\}$ is linearly independent. The case $n=2$ was established by Zhang and Zhao, whereas the case $n\geq 3$ remains open; the conjecture asserts injectivity at the level of the chosen basis representation for the higher-rank quantum unitary group.