The large-characteristic Newton polygon conjecture for twisted binomial L-functions

Let $\chi$ be a multiplicative character of order $c$ over a finite field, let $f(x)=x^d+\lambda x^e$ with $(d,e)=1$, and let $p$ be the characteristic. Write $\operatorname{NP}_{u,m}(f)$ and $\operatorname{NP}_{u,T}(f)$ for the twisted classical and $T$-adic Newton polygons, respectively, and let $P_{u,e,d}$ be the associated lower-bound polygon. Large-characteristic Newton polygon conjecture. If $p$ is large enough with respect to $c$ and $d$, then

$$\operatorname{NP}_{u,m}(f)=\operatorname{NP}_{u,T}(f)=P_{u,e,d}.$$

This conjecture generalizes the conjecture of Zhang and Niu and is motivated by the criterion that equality holds when the associated integral constant is not divisible by $p$. The paper proves the equality for $p>c(d^2-d+1)$ when $e=d-1$, but the general large-characteristic assertion remains open.