The one-dimensional positive sign uncertainty conjecture

Let $\mathbb{A}_+(1)$ denote the optimal constant for the one-dimensional positive sign uncertainty principle, and let

$$\varphi:=\frac{1+\sqrt{5}}2$$

be the golden ratio.

Positive sign uncertainty conjecture.

$$\mathbb{A}_+(1)=(2\varphi)^{-\tfrac12}.$$

The value is supported by numerical evidence from the discrete Fourier-transform problem and would determine the one-dimensional positive sign uncertainty constant exactly. The source presents it as conjectural.