Turbiner's non-separability conjecture for flat Schrödinger operators

A quasi-exactly-solvable or exactly-solvable problem in $\mathbb{R}^2$ is one containing the Laplace–Beltrami operator with a flat-space metric tensor; its variables are non-separable if the problem is not solvable by separation of variables. Turbiner's conjecture. In $\mathbb{R}^2$ there exist no quasi-exactly-solvable or exactly-solvable problems containing the Laplace–Beltrami operator with flat-space metric tensor, which are characterized by non-separable variables. The paper gives a counterexample, so the conjecture is false in general.