Maximal perturbations conjecture

Let $\mathcal{S}_{even}(\mathbb{R})$ denote the space of real-valued even Schwartz functions on $\mathbb{R}$, and let $\widehat{f}$ be the Fourier transform of $f$. For a sequence $(\varepsilon_i)_{i\in\mathbb{N}}$, consider the samples at $0$ and at $\sqrt{i+\varepsilon_i}$ for $i\geq 1$. Maximal perturbations conjecture. There is $\theta>0$ such that, whenever $|\varepsilon_i|<\theta$ for every $i\in\mathbb{N}$, every $f\in\mathcal{S}_{even}(\mathbb{R})$ can be uniquely recovered from

$$f(0),f(\sqrt{1+\varepsilon_1}),f(\sqrt{2+\varepsilon_2}),\dots,$$

together with

$$\widehat{f}(0),\widehat{f}(\sqrt{1+\varepsilon_1}),\widehat{f}(\sqrt{2+\varepsilon_2}),\dots.$$

This would establish recovery under uniformly bounded perturbations of the interpolation nodes; the preceding discussion indicates that proving it requires a new idea beyond the paper's current methods.