Fourier uniqueness-pair conjecture for the square-root sequence

Let $\mathcal{H}_{\operatorname{e}}$ be the function space defined in the paper, and let

$$\Lambda_0=\{\sqrt n:n\ge0\}.$$

A pair $(\Lambda_0,\Lambda_0)$ is a Fourier uniqueness pair for $\mathcal{H}_{\operatorname{e}}$ when the associated Fourier data on these sets determine the relevant function uniquely. Fourier uniqueness-pair conjecture. $(\Lambda_0,\Lambda_0)$ is a Fourier uniqueness pair for $\mathcal{H}_{\operatorname{e}}$. This conjecture is presented as a weaker property that the sequence $\Lambda_0$ may satisfy, since the normalized basis functions do not constitute a Riesz basis for $\mathcal{H}_{\operatorname{e}}$.