  Using light-cone sum rule techniques, we estimate the form factors which
parametrise the hadronic matrix elements that are relevant for semi-leptonic
three-body proton decays. The obtained form factors allow us to determine the
differential rate for the decay of a proton ($p$) into a positron ($e^+$), a
neutral pion ($\pi^0$) and a graviton ($G$), which is the leading proton decay
channel in the effective theory of gravitons and Standard Model particles
(GRSMEFT). The sensitivity of existing and next-generation neutrino experiments
in detecting the $p \to e^+ \pi^0 G$ signature is studied and the
phenomenological implications of our computations for constraints on the
effective mass scale that suppresses the relevant baryon-number violating
GRSMEFT operator are discussed.
