  We investigate the phenomenology of top squarks at the Large Hadron Collider
(LHC) in a supersymmetric model where lepton number is identified with an
approximate $U(1)_R$ symmetry in such a way that one of the left chiral
sneutrinos can acquire a large vacuum expectation value ($vev$) and can play
the role of the down-type Higgs. This $R$-symmetry allows a subset of trilinear
$R$-parity violating interactions, which determine the collider phenomenology
of this model in a significant way. The gauginos are Dirac particles and
gluinos are relatively heavy in this class of models. The model contains a
right handed neutrino superfield, which gives a tree level mass to one of the
active neutrinos. An order one neutrino Yukawa coupling also helps enhance the
Higgs boson mass at the tree level and results in a very light bino-like
neutralino ($\widetilde \chi_2^0$) with mass around a few hundred MeV, which is
a carrier of missing (transverse) energy (\met). The model can accommodate two
rather light top squarks, compatible with the observed mass of the Higgs boson.
The lighter top squark (${\widetilde t}_1$) can decay into
$t\widetilde\chi_2^0$, and thus the signal would be similar to the signal of
top quark pair production at the LHC. In addition, fully visible decays such as
${\widetilde t}_2 \rightarrow b e^+$ can give rise to interesting final states.
Such signals at the LHC combined with other features like a heavy gluino could
provide a strong evidence for this kind of a model. Our analysis shows that
$m_{\widetilde t_1}\lsim 575~(750)$ GeV and $m_{\widetilde t_2}\lsim 1.2~(1.4)$
TeV can be probed with 5$\sigma$ statistical significance at the 13 TeV LHC
with 300~(3000) fb$^{-1}$ of integrated luminosity. Finally, we observe that in
the presence of super-light carriers of \met, the so-called `stealth' top
squark scenario may naturally appear in our model.
