Existence of a long-time limit for the mean phase without rotational invariance

Determine whether the mean phase Θ(t,ω) of the rough Kuramoto model converges as t → ∞ under the non-rotationally invariant setting with identical noise components and a diagonal noise coefficient, specifically under Hypotheses A, B, H_W+, and GII. Prove or disprove the almost sure existence of a limit Θ_∞(ω)=lim_{t→∞}Θ(t,ω) in this regime.

Background

When rotational invariance of G is removed, the authors impose identical components of the driving rough noise (H_W+) and assume G is diagonal and equal on synchronized configurations (H_GII). Under these conditions, they still establish that phases synchronize to a time-dependent reference Θ(t), but they cannot prove the convergence of Θ(t) to a limit as t → ∞.

The unresolved point is whether an almost sure limit Θ∞(ω) exists in this setting; if it does not, the synchronized phases would track a non-convergent random attractor determined by Θ(t), contrasting with the rotationally invariant case where Θ∞(ω) can be characterized.