This sign works equally well for both $q$-Bosons and $t$-Bosons, so the choice of parameter is up to the tourist.
T. Sasamoto and M. Wadati, Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A 31 (1998), 6057–6071
A. Borodin, I. Corwin, L. Petrov, T. Sasamoto, Spectral theory for the q- Boson particle system, Compositio Mathematica 151 (2015), no. 1, 1–67, arXiv:1308.3475 [math-ph]