  A theoretical investigation is done to clarify the origin of the peak
structure observed near the $K^{-} p p$ threshold in the in-flight
${}^{3}\text{He} ( K^{-}, \, \Lambda p ) n$ reaction of the J-PARC E15
experiment, which could be a signal of the lightest kaonic nuclei, that is, the
$\bar{K} N N (I=1/2)$ state. For the investigation, we evaluate the $\Lambda p$
invariant mass spectrum assuming two possible scenarios to interpret the
experimental peak. One assumes that the $\Lambda (1405)$ resonance is generated
after the emission of an energetic neutron from the absorption of the initial
$K^-$, not forming a bound state with the remaining proton. This uncorrelated
$\Lambda (1405) p$ system subsequently decays into the final $\Lambda p$. The
other scenario implies that, after the emission of the energetic neutron, a
$\bar{K} N N$ bound state is formed, decaying eventually into a $\Lambda p$
pair. Our results show that the experimental signal observed in the in-flight
${}^{3}\text{He} ( K^{-} , \, \Lambda p ) n$ reaction at J-PARC is
qualitatively well reproduced by the assumption that a $\bar{K} N N$ bound
state is generated in the reaction, definitely discarding the interpretation in
terms of an uncorrelated $\Lambda (1405) p$ state.
