Strong irreducibility conjecture for the quantum pseudo-Kähler plane representation

Let $\mathcal{C}_{\mathbf{q}}$ and $\mathcal{C}_{1/\mathbf{q}^\vee}$ be the two Hopf algebras represented on the Hilbert space $\mathscr{H}$ by $mathpi$, with the corresponding operator algebras generated by the operators in the representations referred to in the source. An operator is weakly commuting with these algebras when it commutes with all their represented operators on the relevant dense domain. Strong irreducibility conjecture. The operator algebra $mathpi(\mathcal{C}_{1/\mathbf{q}^\vee})$ is the full commutant of $mathpi(\mathcal{C}_{\mathbf{q}})$: every densely defined linear operator on $\mathscr{H}$ that weakly commutes with all operators representing both $\mathcal{C}_{\mathbf{q}}$ and $\mathcal{C}_{1/\mathbf{q}^\vee}$ is a scalar multiple of the identity. This is an analogue of the corresponding irreducibility statement for the modular double quantum plane. The paper does not depend on the conjecture, and no resolution is given.