On supports of characters of the unitriangular group

Let $G$ be a finite group and $\chi$ be its irreducible complex character. The set $\mathrm{Supp}(\chi)=\{g\in G\mid\chi(g)\neq0\}$ is called the support of $\chi$.
Let $G=U$ be the unitriangular group (i.e., the group of unipotent triangular matrices) over a finite field of sufficiently large characteristic. In the paper we introduce the notion of $i$-regular character and describe the support of a $2$-regular character in terms of coefficients of minors of the characteristic matrix.