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In this article, we continue our investigation into the unique continuationproperties of real-valued solutions to elliptic equations in the plane. Moreprecisely, we make another step towards proving a quantitative version ofLandis' conjecture by establishing unique continuation at infinity estimatesfor solutions to equations of the form $- \Delta u + V u = 0$ in$\mathbb{R}^2$, where $V = V_+ - V_-$, $V_+ \in L^\infty$, and $V_-$ is anon-trivial function that exhibits exponential decay at infinity. The main toolin the proof of this theorem is an order of vanishing estimate in combinationwith an iteration scheme. To prove the order of vanishing estimate, weestablish a similarity principle for vector-valued Beltrami systems.

On Landis' conjecture in the plane when the potential has an exponentially decaying negative part