  In this article, we introduce a novel numerical method for determining
optimal Quantum Energy Inequality (QEI) bounds in one- and two-particle states,
extending previous work focused on the one-particle case. Our method,
applicable to integrable models constructed via the inverse scattering
approach, overcomes challenges associated with the numerical treatment of
kinematical singularities and the increased complexity in the two-particle
scenario.
  Applying the method to the sinh-Gordon model provides explicit insights into
optimal QEI bounds. Notably, QEIs persist at the two-particle level, though,
allowing for significantly more negative energy to be sustained over longer
times compared with the one-particle case. Our findings confirm
self-interaction as the source of negative energy, with stronger interactions
yielding more pronounced negativities. Importantly, evidence supports the
finite sustainability of negative energies over time, implying the validity of
the averaged weak energy condition (AWEC) at both one- and two-particle levels.
Last but not least, we identify a one-parameter class of nonminimal stress
tensor expressions satisfying QEIs at both levels, with more stringent
constraints arising from the QEI bounds at two-particle level.
