Asymmetry of a class of Mellin transforms: Request for counterexamples

We study a class of Mellin transforms $\mu_\eta$ associated with functions $\eta$ and investigate the distribution of their zeros in the critical strip. Under suitable assumptions on $\eta$, we obtain a condition preventing $\mu_\eta(s)$ and $\mu_\eta(1-s)$ from vanishing simultaneously away from the critical line when the imaginary part of $s$ is sufficiently large. We formulate a statement and invite the reader to exhibit a counterexample.