  We discuss the unique spin evolution of the young X-ray pulsar PSR
J0537-6910, a system in which the regular spin down is interrupted by glitches
every few months. Drawing on the complete timing data from the Rossi X-ray
Timing Explorer (RXTE, from 1999-2011), we argue that a trend in the
inter-glitch behaviour points to an effective braking index close to $n=7$,
much larger than expected. This value is interesting because it would accord
with the neutron star spinning down due to gravitational waves from an unstable
r-mode. We discuss to what extent this, admittedly speculative, scenario may be
consistent and if the associated gravitational-wave signal would be within
reach of ground based detectors. Our estimates suggest that one may, indeed, be
able to use future observations to test the idea. Further precision timing
would help enhance the achievable sensitivity and we advocate a joint observing
campaign between the Neutron Star Interior Composition ExploreR (NICER) and the
LIGO-Virgo network.
