Rationality conjecture for Mellin transforms of p-adic Whittaker functions

Let $I_{\lambda_0,f}(\lambda)$ denote the Mellin transform associated with the parameters $\lambda_0$, $f$, and $\lambda$, and let $q$ be the residue-field cardinality. Rationality conjecture. The Mellin transform $I_{\lambda_0,f}(\lambda)$ is always a rational function in the variables $q^{\lambda_\alpha}$. The preceding argument establishes rationality in the setting treated there, while the conjecture asserts this for the Mellin transform in general; the supplied text does not provide evidence that the conjecture has been resolved.