Niho's five-value conjecture for crosscorrelation spectra

Let $F$) be a finite field of order $4^m$, where $m$ is even, and set

$$d=4\bigl(\sqrt{|F|}-1\bigr)+1.$$

Let $W_{F,d}(a)$ denote the Weil sum associated with this decimation. Niho's five-value conjecture. The set

$$\{W_{F,d}(a):a\in F^*\}$$

contains at most five distinct values. This conjecture concerns the crosscorrelation spectrum of binary m-sequences for the Niho-type decimation with $s=4$; it was based on numerical results and had remained open for almost half a century, while the paper investigates its resolution.