Does the background spectrum flatten at the lowest frequencies?
In plain words
If black hole pairs are driven together by stars and gas, and not by gravitational waves alone, at wide separations, the background should be weaker at the longest periods. Measuring where this happens tells how binaries cross the last parsec.
Precise statement
Measure the slope of $h_c(f)$ below $f \sim 3\ \mathrm{nHz}$ and determine whether it departs from the pure gravitational-wave-driven power law $h_c \sim f^{(-2/3)}$, and at what turnover frequency $f_t$. Environmental hardening by stellar scattering or circumbinary gas, or high orbital eccentricity, predicts a turnover whose location constrains stellar densities and eccentricities in merger remnants.
What would settle it
Pulsar timing baselines of 20 years or more with low-frequency noise modeling sufficient to measure $h_c$ at $f \sim 1-3\,\mathrm{nHz}$.