The eigenvalue conjecture for Radon transforms on flag-quotient functions

Let $\lambda$ be fixed, let $U_{\lambda}$ be the subgroup associated with $\lambda$, and let the Radon transform $\operatorname{\mathcal{R}}$ act on the complex-valued function space $\mathbb{C}[GL_n(\mathbb{F}_q)/U_{\lambda}]$. Radon transform eigenvalue conjecture. The eigenvalues of $\operatorname{\mathcal{R}}$ are powers of $q$. The preceding computations establish this pattern in the cases examined, but the statement is presented as a conjecture for fixed $\lambda$ in general.