  Super-Eddington accretion of neutron stars (NSs) has been suggested both
observationally and theoretically. In this paper, we propose that NSs in
close-orbit binary systems with companions of helium (He) stars, most of which
systems form after the common-envelope phase, could experience super-Eddington
stable Case BB/BC mass transfer (MT), and can sometimes occur accretion-induced
collapses (AICs) to form lower mass-gap black holes (mgBHs). Our detailed
binary evolution simulations reveal that AIC events tend to happen if the
primaries NS have an initial mass $\gtrsim1.7\,M_\odot$ with an accretion rate
of $\gtrsim300$ times the Eddington limit. These mgBHs would have a mass nearly
equal to or slightly higher than the NS maximum mass. The remnant mgBH--NS
binaries after the core collapses of He stars are potential progenitors of
gravitational-wave (GW) source. Multimessenger observation between GW and
kilonova signals from a population of high-mass binary NS and mgBH--NS mergers
formed through super-Eddington stable MT are helpful in constraining the
maximum mass and equation of state of NSs. S230529ay, a mgBH--NS merger
candidate recently detected in the fourth observing run of the LIGO-Virgo-KAGRA
Collaboration, could possibly originate from this formation scenario.
