Priorities in gravitational waveforms for future space-borne detectors: vacuum accuracy or environment?

In preparation for future space-borne gravitational-wave (GW) detectors, should the modelling effort focus on high-precision vacuum templates or on the astrophysical environment of the sources? We perform a systematic comparison of the phase contributions caused by 1) known environmental effects in both gaseous and stellar matter backgrounds, or 2) high-order post-Newtonian {(PN)} terms in the evolution of mHz GW sources {during the inspiral stage of massive binaries}. We use the accuracy of currently available analytical waveform models as a benchmark {value, finding} the following trends: the largest unmodelled phase contributions are likely environmental rather than PN for binaries lighter than $\sim 10^7/(1+z)^2$~M$_{\odot}$, where $z$ is the redshift. Binaries heavier than $\sim 10^8/(1+z)$~M$_{\odot}$ do not require more accurate {inspiral} waveforms due to low signal-to-noise ratios (SNRs). For high-SNR sources, environmental {phase contributions} are relevant at low redshift, while high-order vacuum templates are required at $z > 4$. Led by these findings, we argue that including environmental effects in waveform models should be prioritised in order to maximize the science yield of future mHz detectors.