  We study the potential of the LHC accelerator, and a future 33 TeV proton
collider, to observe the production of a light top squark pair in association
with the lightest Higgs boson (${\tilde t}_1 {\tilde t}_1 h_1$), as predicted
by the Next-to-Minimal Supersymmetric Standard Model (NMSSM). We scan randomly
about ten million points of the NMSSM parameter space, allowing all possible
decays of the lightest top squark and lightest Higgs boson, with no assumptions
about their decay rates, except for known physical constraints such as
perturbative bounds, Dark matter relic density consistent with recent Planck
experiment measurements, Higgs mass bounds on the next to lightest Higgs boson,
$h_2$, assuming it is consistent with LHC measurements for the Standard Model
Higgs boson, LEP bounds for the chargino mass and Z invisible width,
experimental bounds on B meson rare decays and some LHC experimental bounds on
SUSY particle spectra different to the particles involved in our study. We find
that for low mass top-squark, the dominating decay mode is ${\tilde t}_1
\rightarrow b \tilde{\chi}_1^{\pm}$ with $\tilde{\chi}_1^{\pm} \rightarrow
W^{\pm} \tilde{\chi}_1^0 $. We use three bench mark points with the highest
cross sections, which naturally fall within the compressed spectra of the top
squark, and make a phenomenological analysis to determine the optimal event
selection that maximizes the signal significance over backgrounds. We focus on
the leptonic decays of both $W$'s and the decay of lightest Higgs boson into
b-quarks ($h_1 \rightarrow b \bar b$). Our results show that the high
luminosity LHC will have limitations to observe the studied signal and only a
proton collider with higher energy will be able to observe the SUSY scenario
studied with more than three standard deviations over background.
