  Gravitational waves (GWs) in the nano-hertz band are great tools for
understanding the cosmological evolution of supermassive black holes (SMBHs) in
galactic nuclei. We consider SMBH binaries in high-$z$ ultra-luminous infrared
galaxies (ULIRGs) as sources of a stochastic GW background (GWB). ULIRGs are
likely associated with gas-rich galaxy mergers containing SMBHs that possibly
occur at most once in the life of galaxies, unlike multiple dry mergers at low
redshift. Adopting a well-established sample of ULIRGs, we study the properties
of the GWB due to coalescing binary SMBHs in these galaxies. Since the ULIRG
population peaks at $z>1.5$, the amplitude of the GWB is not affected even if
BH mergers are delayed by as long as $\sim $ 10 Gyrs. Despite the rarity of the
high-$z$ ULIRGs, we find a tension with the upper limits from Pulsar Timing
Array (PTA) experiments. This result suggests that if a fraction $f_{\rm
m,gal}$ of ULIRGs are associated with SMBH binaries, then no more than $20
f_{\rm m,gal}(\lambda_{\rm Edd}/0.3)^{5/3}(t_{\rm life}/30~{\rm Myr})~\%$ of
the binary SMBHs in ULIRGs can merge within a Hubble time, for plausible values
of the Eddington ratio of ULIRGs ($\lambda_{\rm Edd}$) and their lifetime
($t_{\rm life}$).
