  In gravitational-wave astronomy, extreme-mass-ratio-inspiral (EMRI) sources
for the upcoming LISA observatory have the potential to serve as high-precision
probes of astrophysical environments in galactic nuclei, and of potential
deviations from general relativity (GR). Such ``beyond vacuum-GR'' effects are
often modeled as perturbations to the evolution of vacuum EMRIs under GR.
Previous studies have reported unprecedented constraints on these effects by
examining the inference of one effect at a time. However, a more realistic
analysis would require the simultaneous inference of multiple beyond vacuum-GR
effects. The parameters describing such effects are generally significantly
correlated with each other and the vacuum EMRI parameters. We explicitly show
how these correlations remain even if any modeled effect is absent in the
actual signal, and how they cause inference bias when any effect in the signal
is absent in the analysis model. This worsens the overall measurability of the
whole parameter set, challenging the constraints found by previous studies, and
posing a general problem for the modeling and inference of beyond vacuum-GR
effects in EMRIs.
