  We report detailed X-ray observations of the unique binary system $\epsilon$
Lupi, the only known short-period binary consisting of two magnetic early-type
stars. The components have comparably strong, but anti-aligned magnetic fields.
The orbital and magnetic properties of the system imply that the magnetospheres
overlap at all orbital phases, suggesting the possibility of variable
inter-star magnetospheric interaction due to the non-negligible eccentricity of
the orbit. To investigate this effect, we observed the X-ray emission from
$\epsilon$ Lupi both near and away from periastron passage, using the Neutron
Star Interior Composition Explorer mission (NICER) X-ray Telescope. We find
that the system produces excess X-ray emission at the periastron phase,
suggesting the presence of variable inter-star magnetospheric interaction. We
also discover that the enhancement at periastron is confined to a very narrow
orbital phase range ($\approx 5\%$ of the orbital period), but the X-ray
properties close to periastron phase are similar to those observed away from
periastron. From these observations, we infer that the underlying cause is
magnetic reconnection heating the stellar wind plasma, rather than shocks
produced by wind-wind collision. Finally, by comparing the behavior of
$\epsilon$ Lupi with that observed for cooler magnetic binary systems, we
propose that elevated X-ray flux at periastron phase is likely a general
characteristic of interacting magnetospheres irrespective of the spectral types
of the constituent stars.
