Gupta–Rai conjecture on permutation trinomials over finite fields

Let $q=p^k$, where $p>7$ is a prime and $k>1$, and let $ff$ be the finite field with $q$ elements. For $\alpha\inff^*$, define

$$f(X)=X^{q(p-1)+1}+\alpha X^{pq}+X^{q+p-1}.$$

Gupta–Rai conjecture. The trinomial $f(X)$ is a permutation polynomial over $\mathbb F_{q^2}$ if and only if $\alpha=-1$ and $k=2$. This conjecture concerns the permutation behaviour of a family of trinomials over quadratic extensions of finite fields. Earlier work establishes related classifications in characteristics $3$, $5$, and $7$, while the case $p>7$ and $k>1$ is the proposed general case.