Equality of Kirillov functions and irreducible characters for unitriangular groups

Let $q$ be a prime power, let $n\leq 12$, and let $\mathfrak{u}_n(q)^*$ be the dual of the Lie algebra of strictly upper-triangular $n\times n$ matrices over the finite field with $q$ elements. For $\lambda\in\mathfrak{u}_n(q)^*$, write $\psi_\lambda$ for the associated irreducible character, $\psi^{\mathrm{Exp}}_\lambda$ for the character obtained using the exponential construction, and $\xi_\lambda$ for the Kirillov function.

The conjecture. For every $\lambda\in\mathfrak{u}_n(q)^*$,

$$\psi_\lambda=\psi^{\mathrm{Exp}}_\lambda=\xi_\lambda\in\operatorname{Irr}(\mathrm{UT}_n(q)).$$

This would show that, for $n\leq 12$, every irreducible character of the unitriangular group is obtained directly from a functional by the Kirillov construction. The statement is known for $q=2$ from computer calculations, but the source presents it as conjectural for general prime powers and suggests that a verification analogous to Evseev's computational work may be possible.