Relating LHC event rates to supersymmetric Grand Unified Theories containing $SU(5)$

We elaborate on a recently found $SU(5)$ relation confined to the up-(s)quark flavour space, that remains immune to large quantum corrections up to the TeV scale. We investigate the possibilities opened by this new window on the GUT scale in order to find TeV-scale $SU(5)$ tests realizable at the LHC. We present a variety of tests, which appear as relations among observables involving flavour violation or chirality flips and rely on the techniques of top polarimetry, charm-tagging, or Higgs detection from cascade decays. We discuss the cases of natural Supersymmetry and top-charm Supersymmetry as example cases. We find that $O(10)$ to $O(100)$ events are needed to obtain 50\% of relative precision at 3$\sigma$ significance for all proposed tests.