  We analyze the prospects for observing the CP-even neutral Higgs bosons (h
and H) in their decays into flavor violating $b \bar s+ c.c.$ channels at the
proposed LHeC, with $\sqrt s \approx 1.3$ TeV, in the 2HDM Type-III, assuming a
four-zero texture in the Yukawa matrices and a general Higgs potential. We
consider theoretically consistent scenarios in agreement with current
experimental data from flavor and Higgs physics. We investigate the charged
current production process $\nu_e \phi q$ in presence of flavor violating
decays of the Higgs bosons, that lead to a 3-jets + $E\!\!\!\!/_T$ signature.
We demand exactly two jets, one tagged $b$-jet and one light-flavor jet, all in
the central rapidity region. The remaining jet (originated by the remanent
quark $q$) is tagged in the forward or backward regions and this together with
a central jet veto (not more than one light-flavor jet) are essential
criterions to enhance the signal--to--background rates. We consider the most
relevant Standard Model (SM) backgrounds, treating $c$-jets separately from
light-flavor and gluon ones, while allowing for mis-tagging. We find that the
SM-like Higgs boson, $h$, would be accessible within several parameter
configurations of our model at approximately the 1-2$\sigma$ level with 100
fb$^{-1}$ of data. We also find that the heaviest neutral Higgs boson, $H$,
with mass up to 150 GeV, would have a 1$\sigma$ significance for the same data
sample. At the end of the LHeC running, one would have ten times data
accumulation and for all the Higgs masses the significances are enhanced so as
to allow for detection of both the $h$ and $H$ state. Hence, one of the most
viable extensions of 2HDMs with FCNC's generated at tree level but controlled
by a four-zero texture approach in the Yukawa matrices, as opposed to the
adoption of ad hoc discrete symmetries, could be put under scrutiny at a future
LHeC.
