Environmental effects vs. modified gravity in the LISA massive black hole binary population

Gravitational-wave signals from massive black hole binaries observed by LISA can carry imprints of both the astrophysical environment of the source and possible deviations from general relativity. We investigate whether environmental effects leave a detectable imprint on the LISA binary population, and whether they can mimic modified-gravity effects with the same frequency dependence. As representative channels we adopt accretion and viscous migration in a circumbinary disk for the environmental sector, and a time-varying Newton constant $\dot G$ for the modified-gravity sector. All three effects enter the waveform at the same negative post-Newtonian order and are described, at leading order, by a common phase-deformation parameter, which makes them formally degenerate at the single-event level. Combining Fisher-matrix forecasts with a hierarchical nested-sampling analysis of synthetic catalogs from astrophysically motivated population models, we find that, even under extreme astrophysical assumptions -- an active fraction of $50\%$, together with a super-Eddington accretion tail -- the population-level posteriors remain fully compatible with vacuum. However, a hierarchical population-wide analysis may yield a non-trivial upper limit on the active fraction and a mild lower bound on the slope of the Eddington-ratio distribution. Environmental effects are therefore unlikely to bias LISA's tests of general relativity with massive black hole binaries in astrophysically realistic scenarios.