  The interpretation of LHC data, and the assessment of possible hints of new
physics, require the precise knowledge of the proton structure in terms of
parton distribution functions (PDFs). We present a systematic methodology
designed to determine whether and how global PDF fits might inadvertently 'fit
away' signs of new physics in the high-energy tails of the distributions. We
showcase a scenario for the High-Luminosity LHC, in which the PDFs may
completely absorb such signs of new physics, thus biasing theoretical
predictions and interpretations. We discuss strategies to single out the
effects in this scenario, and disentangle the inconsistencies that stem from
them. Our study brings to light the synergy between the high luminosity
programme at the LHC and future low-energy non-LHC measurements of large-$x$
sea quark distributions. The analysis code used in this work is made public so
that any users can test the robustness of the signal associated to a given BSM
model against absorption by the PDFs.
