Nonnegative plethystic characteristic conjecture for induced supercharacters of unitriangular groups

Let $\mathbb F_q$ be a fixed finite field, let $U_n$ be the group of unipotent upper-triangular matrices over $\mathbb F_q$, let $G_n=GL_n(\mathbb F_q)$, and let $\chi$ be a supercharacter of $U_n$. Write $\chi\uparrow_{U_n}^{G_n}$ for the character induced to $G_n$, and let $\rho\circ ch$ denote the characteristic-map construction used in the paper, with $\rho_1,\ldots,\rho_n$ its character generators. Nonnegative plethystic characteristic conjecture. We have

$$\rho\circ ch(\chi\uparrow_{U_n}^{G_n})\in\mathbb N[\rho_1,\ldots,\rho_n].$$

The conjecture asserts that the characteristic of every character induced from a supercharacter of $U_n$ has nonnegative integral coefficients in the generators $\rho_1,\ldots,\rho_n$. The paper provides computations for small examples, including cases with $q=2$, but the general assertion is left open.