Pulsar timing array collaborations have recently reported evidence for a noise process with a common spectrum among the millisecond pulsars in the arrays. The spectral properties of this common-noise process are consistent with expectations for an isotropic gravitational-wave background (GWB) from inspiralling supermassive black-hole binaries. However, recent simulation analyses based on Parkes Pulsar Timing Array data indicate that such a detection may arise spuriously. In this paper, we use simulated pulsar timing array datasets to further test the robustness of the inference methods for spectral and spatial correlations from a GWB. Expanding on our previous results, we find strong support (Bayes factors exceeding 10 5 ) for the presence of a common-spectrum noise process in datasets where no common process is present, under a wide range of timing noise prescriptions per pulsar. We show that these results are highly sensitive to the choice of Bayesian priors on timing noise parameters, with priors that more closely match the injected distributions of timing noise parameters resulting in diminished support for a common-spectrum noise process. These results emphasize shortcomings in current methods for inferring the presence of a common-spectrum process, and imply that the detection of a common process is not a reliable precursor to detection of the GWB. Future searches for the nanohertz GWB should remain focussed on detecting spatial correlations, and make use of more tailored specifications for a common-spectrum noise process.