  We present a search for continuous gravitational-wave emission due to r-modes
in the pulsar PSR J0537-6910 using data from the LIGO-Virgo Collaboration
observing run O3. PSR J0537-6910 is a young energetic X-ray pulsar and is the
most frequent glitcher known. The inter-glitch braking index of the pulsar
suggests that gravitational-wave emission due to r-mode oscillations may play
an important role in the spin evolution of this pulsar. Theoretical models
confirm this possibility and predict emission at a level that can be probed by
ground-based detectors. In order to explore this scenario, we search for r-mode
emission in the epochs between glitches by using a contemporaneous timing
ephemeris obtained from NICER data. We do not detect any signals in the
theoretically expected band of 86-97 Hz, and report upper limits on the
amplitude of the gravitational waves. Our results improve on previous amplitude
upper limits from r-modes in J0537-6910 by a factor of up to 3 and place
stringent constraints on theoretical models for r-mode driven spin-down in PSR
J0537-6910, especially for higher frequencies at which our results reach below
the spin-down limit defined by energy conservation.
