  Electromagnetic observations reveal that almost all galaxies have
supermassive black holes (SMBHs) at their centers, but their properties,
especially their spins, are not fully understood. Some of the authors have
recently shown [Oshita and Tsuna (2023)] that rapid spins of $>0.9$, inferred
for masses around $10^7\ M_\odot$ from observations of local SMBHs and
cosmological simulations, source {\it long-lived} ringdowns that enhance the
precision of black hole spectroscopy to test gravity in the near-extreme Kerr
spacetime. In this work, we estimate the statistical errors in the SMBH
mass-spin inference in anticipation of the LISA's detection of extreme
mass-ratio mergers. We show that for rapidly spinning SMBHs, more precise mass
and spin measurements are expected due to the excitations of higher angular
modes. For a near-extremal SMBH of mass $10^7M_\odot$ merging with a smaller BH
with mass ratio $10^{-3}$ at a luminosity distance of $\lesssim
10\:\mathrm{Gpc}$ (redshift $z \lesssim 1.37$), the measurement errors in the
mass and spin of the SMBH would be $\sim 1\:\mathrm{\%}$ and $\sim
10^{-1}\:\mathrm{\%}$ respectively.
