Five-valued conjecture for Weil sums with Niho exponents

Let $p$ be an odd prime, let $F$ be a finite field of order $p^n$, and let $L$ be a quadratic extension of $F$. Let $s=1+k(p^n-1)$ be an invertible Niho exponent over $L$, and define

$$d_1=\gcd(k,p^n+1),\qquad d_2=\gcd(k-1,p^n+1).$$

Five-valued conjecture. If either $d_1+d_2\geq 5$, or $d_1+d_2=3$ and $p^n\equiv 11\pmod{12}$, then the Weil spectrum over $L$ is at least five-valued. In the first case its five values are

$$\{0,-p^n,p^n,2\alpha p^n,(2\beta+1)p^n\},$$

where $lpha,\beta\geq 1$ are integers; in the second case at least four values are

$$\{0,-p^n,p^n,2p^n\}.$$

The paper presents this as a conjecture based on numerical evidence, concerning the possible value distribution of Weil sums for Niho exponents. The supplied text does not give evidence of a resolution.