Sign conjecture for Hardy inequalities in globally twisted waveguides

Let $\Omega$ be a globally twisted waveguide with twist derivative $\dot\theta=\beta+\varepsilon$, where $\beta$ is the constant twisting rate and $\varepsilon$ is a nonzero perturbation vanishing at infinity. Let $-\Delta_D^{\Omega}$ be the Dirichlet Laplacian and let $\lambda_1$ be the lowest eigenvalue of the transverse operator $-\Delta_D^\omega-\beta^2\partial_\tau^2$. A Hardy inequality is an estimate of the form

$$-\Delta_D^{\Omega}-\lambda_1\geq\rho(\cdot),$$

with a non-trivial function $\rho:\mathbb{R}\to[0,\infty)$. Sign conjecture. The Hardy inequality holds if $\beta\varepsilon$ is non-negative and $\varepsilon$ is not identically equal to zero. This would exclude discrete eigenvalues below the essential-spectrum threshold $\lambda_1$ in this regime, opposite to the locally slowed-down twisting regime.