  The recent evidence of a stochastic background of gravitational waves in the
nHz band by pulsar-timing array (PTA) experiments has shed new light on the
formation and evolution of massive black hole binaries with masses $\sim
10^8$--$10^9 M_\odot$. The PTA data are consistent with a population of such
binaries merging efficiently after the coalescence of their galactic hosts, and
presenting masses slightly larger than previously expected. This momentous
discovery calls for investigating the prospects of detecting the smaller ($\sim
10^5$--$10^7 M_\odot$) massive black hole binaries targeted by the Laser
Interferometer Space Antenna (LISA). By using semi-analytic models for the
formation and evolution of massive black hole binaries calibrated against the
PTA results, we find that LISA will observe at least a dozen and up to
thousands of black hole binaries during its mission duration. The minimum
number of detections rises to $\sim 70$ if one excludes models that only
marginally reproduce the quasar luminosity function at $z=6$. We also assess
LISA's parameter estimation capabilities with state-of-the-art waveforms
including higher modes and realistic instrumental response, and find that the
masses, sky position, and distance will typically be estimated to within
respectively 1%, 10 square degrees, and 10% for the detected systems (assuming
a 4-year mission).
