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We continue our investigation of kinetic models of a one-dimensional gas incontact with homogeneous thermal reservoirs at different temperatures.Nonlinear collisional interactions between particles are modeled by a so-calledBGK dynamics which conserves local energy and particle density. Weighting thenonlinear BGK term with a parameter $\alpha\in [0,1]$, and thelinearinteraction with the reservoirs by $(1-\alpha)$, we prove that for all$\alpha$ close enough to zero, the explicit spatially uniform non-equilibriumstable state (NESS) is \emph{unique}, and there are no spatially non-uniformNESS with a spatial density $\rho$ belonging to $L^p$ for any $p>1$. We alsoshow that for all $\alpha\in [0,1]$, the spatially uniform NESS is dynamicallystable, with small perturbation converging to zero exponentially fast.

Uniqueness of the Non-Equilibrium Steady State for a $1$d BGK model in kinetic theory