  Inspiraling massive black-hole binaries (MBHBs) forming in the aftermath of
galaxy mergers are expected to be the loudest gravitational-wave (GW) sources
relevant for pulsar-timing arrays (PTAs) at nHz frequencies. The incoherent
overlap of signals from a cosmic population of MBHBs gives rise to a stochastic
GW background (GWB) with characteristic strain around $h_c\sim10^{-15}$ at a
reference frequency of 1 yr$^{-1}$, although uncertainties around this value
are large. Current PTAs are piercing into the GW amplitude range predicted by
MBHB-population models, but no detection has been reported so far. To assess
the future success prospects of PTA experiments, it is therefore important to
estimate the minimum GWB level consistent with our current understanding of the
formation and evolution of galaxies and massive black holes (MBHs). To this
purpose, we couple a semianalytic model of galaxy evolution and an extensive
study of the statistical outcome of triple MBH interactions. We show that even
in the most pessimistic scenario where all MBHBs stall before entering the
GW-dominated regime, triple interactions resulting from subsequent galaxy
mergers inevitably drive a considerable fraction of the MBHB population to
coalescence. At frequencies relevant for PTA, the resulting GWB is only a
factor of 2-to-3 suppressed compared to a fiducial model where binaries are
allowed to merge over Gyr timescales. Coupled with current estimates of the
expected GWB amplitude range, our findings suggest that the minimum GWB from
cosmic MBHBs is unlikely to be lower than $h_c\sim 10^{-16}$ (at $f = 1$
yr$^{-1}$), well within the expected sensitivity of projected PTAs based on
future observations with FAST, MeerKAT and SKA.
