  A link between magnetars and fast radio burst (FRB) sources has finally been
established. In this context, one of the open issues is whether/which sources
of extra galactic FRBs exhibit X/gamma-ray outbursts and whether it is
correlated with radio activity. We aim to constrain possible X/gamma-ray burst
activity from one of the nearest extragalactic FRB sources currently known over
a broad energy range, by looking for bursts over a range of timescales and
energies that are compatible with being powerful flares from extragalactic
magnetars. We followed up the as-yet nearest extragalactic FRB source at a mere
149 Mpc distance, the periodic repeater FRB180916.J0158+65, during the active
phase on February 4-7, 2020, with the Insight-Hard X-ray Modulation Telescope
(HXMT). Taking advantage of the combination of broad band, large effective
area, and several independent detectors available, we searched for bursts over
a set of timescales from 1 ms to 1.024 s with a sensitive algorithm, that had
previously been characterised and optimised. Moreover, through simulations we
studied the sensitivity of our technique in the released energy-duration phase
space for a set of synthetic flares and assuming different energy spectra. We
constrain the possible occurrence of flares in the 1-100 keV energy band to
E<10^46 erg for durations <0.1 s over several tens of ks exposure. We can rule
out the occurrence of giant flares similar to the ones that were observed in
the few cases of Galactic magnetars. The absence of reported radio activity
during our observations does not allow us to make any statements on the
possible simultaneous high-energy emission.
