Landis's conjecture for exponentially decaying Schrödinger solutions

Let $\Omega$ be either $\mathbb{R}^N$ or an exterior domain in $\mathbb{R}^N$, and let $V\in L^{\infty}(\mathbb{R}^N)$ satisfy $\|V\|_{\infty}\leq 1$. Suppose that $u$ solves

$$\Delta u+Vu=0\quad\text{in }\Omega$$

and that $u(x)=O(e^{-\kappa|x|})$ as $|x|\to\infty$ for some $\kappa>1$. Landis's conjecture. Then $u\equiv 0$ in $\Omega$. The conjecture asks whether sufficiently rapid exponential decay forces unique continuation at infinity for solutions of a linear Schrödinger equation. It was negatively answered by Meshkov, who constructed a complex-valued potential and a nontrivial solution with decay $O(e^{-\kappa|x|^{4/3}})$ at infinity for some $\kappa>0$.