  Interstellar neutral (ISN) hydrogen is the most abundant species in the outer
heliosheath and the very local interstellar medium (VLISM). Charge exchange
collisions in the outer heliosheath result in filtration, reducing the ISN
hydrogen density inside the heliosphere. Additionally, these atoms are
intensively ionized close to the Sun, resulting in a substantial reduction of
their density within a few au from the Sun. The products of this ionization -
pickup ions (PUIs) - are detected by charged particle detectors. The Solar Wind
Around Pluto (SWAP) instrument on New Horizons provides, for the first time,
PUI observations from the distant heliosphere. We analyze the observations
collected between 22 and 52 au from the Sun to find the ISN hydrogen density
profile and compare the results with predictions from global heliosphere
models. We conclude that the density profile derived from the observations is
inconsistent with steady-state model predictions. This discrepancy is not
explained by time variations close to the Sun and thus may be related to the
temporal evolution of the outer boundaries or VLISM conditions. Furthermore, we
show that the cold and hot models of ISN hydrogen distribution are not a good
approximation closer to the termination shock. Therefore, we recommend a new
fiduciary point based on the available New Horizons observations at 40 au from
the Sun, at ecliptic direction (285.62{\deg}, 1.94{\deg}), where the ISN
hydrogen density is 0.11 cm$^{-3}$. The continued operation of New Horizons
should give better insight into the source of the discussed discrepancy.
