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Massive black hole binaries (MBHBs) could be the loudest gravitational-wave(GW) sources in milli-Hertz (mHz) GW band, but their dynamical evolution maystall when the black holes reach the innermost parsec of a galaxy. Such a"final-parsec problem" could be solved if MBHB forms in a gas-rich environment,such as an active galactic nucleus (AGN), but other solutions not involvingAGNs also exist. Testing the correlation between these mHz GW sources and AGNsis difficult in real observation because AGNs are ubiquitous. To overcome thisdifficult, we use a statistical method, first designed to constrain the hostgalaxies of stellar-mass binary black holes, to search for a MBHB-AGNcorrelation in different astrophysical scenarios. We find that by detectingonly one MBHB at $z \lesssim 0.5$, a mHz GW detector, such as the LaserInterferometer Space Antenna (LISA), can already distinguish different mergerscenarios thanks to the precise localization of the source. Future detectornetworks and deeper AGNs surveys can further testify the MBHB-AGN correlationup to a redshift of $z\sim 2$ even if only a small fraction of MBHBs mergeinside AGNs. These constraints will help settle the long-standing debate on thepossible solutions to the final-parsec problem.

Unveiling the solution to the final-parsec problem by combining milli-Hertz gravitational-wave observation and AGN survey