  The long-standing discrepancy between the experimental determination by the
Muon $g-2$ Collaboration at Brookhaven and the Standard Model predictions for
the anomalous magnetic moment of the muon cannot be explained within simple
unified framework like the Constrained Minimal Supersymmetric Standard Model,
but it can within its extension with vector-like fermions. In this paper we
consider a model with an additional vector-like $5+\bar{5}$ pair of $SU(5)$.
Within this model we first identify its parameter space that is consistent with
the current discrepancy and show that this implies the lighter chargino mass in
the range of $700-1200$ GeV. We examine how it is affected by constraints from
electroweak sparticle search at the LHC based on 13 TeV search with 36.1
${fb}^{-1}$ integrated luminosity. We show that null trilepton signal searches
coming from chargino-neutralino pair production significantly constrains the
allowed parameter space except when the chargino-neutralino mass difference is
relatively small, below about 10 GeV. Next we consider the expected impact of
the New Muon $g-2$ experiment at Fermilab with its projected sensitivity reach
of $7\,\sigma$ and, assuming it confirms the current discrepancy, show that the
remaining parameter space of the considered model will be in strong tension
with the current LHC limits.
