Landis's qualitative conjecture for Schrödinger equations

For a potential $V \in L^{\infty}(\mathbb{R}^N)$, let $u$ be a solution to the Schrödinger elliptic equation

$$- \Delta u + V(x)u = 0 \quad \text{in } \mathbb{R}^N.$$

Suppose that

$$\exists C,\varepsilon>0,\ \forall x\in\mathbb{R}^N,\quad |u(x)|\leq C\exp(-|x|^{1+\varepsilon}).$$

“Qualitative” Landis conjecture. Then $u\equiv 0$.

This conjecture asserts unique continuation from sufficiently rapid decay at infinity for Schrödinger equations with bounded potentials. The source presents it as the motivation for its dual gradient estimates; its resolution status is not specified in the supplied text.