Continuity conjecture for Kuksin measures

Let $\boldsymbol{a}$ denote the parameters defining the stochastic forcing, and let Kuksin measures be the statistically steady measures associated with the stochastic Navier–Stokes equations on $\mathbb{T}^2$. The vorticities are already known to lie in $H^1(\mathbb{T}^2)\cap L^\infty(\mathbb{T}^2)$.

Continuity conjecture. Kuksin measures are in fact supported on continuous vorticities.

This conjecture is motivated by Statistical Mechanics predictions for the end states of the two-dimensional Euler dynamics. The paper does not establish continuity, and the conjecture remains open.