Theoretical interpretation of CHSH correlation loss: Classical phase-space thermal noise vs. non-unitary decoherence

I am examining the theoretical bounds of CHSH correlation degradation under a deterministic phase-space framework (in the spirit of local-realist / hidden-variable foundational models).

Consider an empirical system yielding a CHSH value of $S = 2.404$ (via single correlations $E_1 = +0.556$, $E_2 = -0.672$, $E_3 = +0.592$, $E_4 = +0.584$), which sits strictly between the local classical limit ($S \le 2.0$) and the maximal Tsirelson bound ($2\sqrt{2} \approx 2.828$).

Assuming the underlying state space dynamics are deterministic and governed by continuous trajectory equations:

From a statistical mechanics perspective, can the intermediate value $S = 2.404$ be rigorously modeled purely as classical thermal fluctuations (e.g., Liouville/Langevin noise) over a deterministic phase space, or does any reduction below $2\sqrt{2} \approx 2.828$ fundamentally necessitate non-unitary open quantum system dynamics (such as explicit operator-level $T_1/T_2$ phase damping and amplitude damping channels)?

(CHSH stands for Clauser–Horne–Shimony–Holt. It is one of the most well-known tools in quantum entanglement and Bell tests.)