Conjecture on exponential Kirillov functions for unitriangular groups

Let $p>0$ be the characteristic of $\mathbb{F}_q$, let $G=\mathrm{UT}_n(q)$, and let $\mathfrak{u}_n(q)$ be the algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{F}_q$. For a Kirillov function $\psi$, define its exponential version using the truncated exponential map

$$\operatorname{Exp}(X)=1+X+\frac{1}{2}X^2+\dots+\frac{1}{(p-1)!}X^{p-1},$$

by

$$\psi^{\operatorname{Exp}}(\operatorname{Exp}(X))=\psi(1+X).$$

Exponential Kirillov-function conjecture. The irreducible characters and exponential Kirillov functions of $\mathrm{UT}_n(q)$ coincide if and only if $n\leq 6p$.

The claim strengthens the known range $n<2p$ and is consistent with the paper's construction of an exponential Kirillov function of degree $q^{5p^2-p}$ that is not a character when $n>6p$.