  The existence of a primordial stochastic gravitational wave background (SGWB)
is a common prediction in various models of the early Universe. Despite
constraints at different frequency ranges and claims of detection in the nHz
range by Pulsar Timing Arrays, the amplitude and spectral dependence of the
SGWB in the mHz range remain largely unknown. Plausible models of early
Universe Physics predict a wide range of SGWB amplitudes, from undetectable to
exceeding the constraints from Big Bang Nucleosynthesis. This paper explores
the potential impact of a prominent primordial SGWB on LISA's main scientific
targets. Our main analyses focuses on Massive Black Hole Binaries (MBHBs). By
employing publicly available MBHB population models and state-of-the-art LISA's
forecasting pipeline, we analyze the effects of the SGWB on MBHB detections. We
find that the decrease in the signal-to-noise ratio induced by a strong
primordial GWB can significantly reduce the number of detectable events,
compromise the precision of constraints, and even hinder sky localization for
some events. We also examine the impact of the SGWB on Stellar Origin Black
Hole Binaries (SOBHBs) and Galactic Binaries (GBs), which are fainter sources
than MBHBs. Depending on the spectral properties of the SGWB, we conclude that
these sources could be either marginally affected or rendered completely
undetectable. This largely unexplored aspect raises critical questions about
the potential challenges posed by a prominent SGWB to LISA's astrophysical
objectives, including MBHBs, SOBHBs, and GBs.
