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Chandra observations of the nearby candidate Lyman continuum (LyC) emitting galaxy Tol 0440-381 show brightening of an X-ray source by at least a factor of 4 to a luminosity of |$1.6 \times 10^{40} \rm \, erg \, s^{-1}$| over 3.8 d. The X-ray emission likely arises from either a low-luminosity active galactic nucleus or an ultraluminous X-ray source. The properties of the X-ray source are similar to those found in Haro 11 and Tololo 1247-232, the only other LyC emitting galaxies that have been resolved in X-rays. All three galaxies host luminous, variable, and hard-spectrum X-ray sources that are likely accretion-powered. Accretion on to compact objects produces powerful outflows and ionizing radiation that could help enable LyC escape. [ABSTRACT FROM AUTHOR]