  The three-particle $K$-matrix, $\mathcal{K}_{\mathrm{df},3}$, is a
scheme-dependent quantity that parametrizes short-range three-particle
interactions in the relativistic-field-theory three-particle finite-volume
formalism. In this work, we compute its value for systems of three pions at
maximal isospin through next-to-leading order (NLO) in Chiral Perturbation
Theory (ChPT). We compare the values to existing lattice QCD results and find
that the agreement between lattice QCD data and ChPT in the first two
coefficients of the threshold expansion of $\mathcal{K}_{\mathrm{df},3}$ is
significantly improved with respect to leading order once NLO effects are
incorporated.
