Conjecture on scaling limits selecting viscosity solutions of Euler equations

Let $Q_{N_i,\nu_i}$ be probability laws associated with sequences $\nu_i\to 0$ and $N_i\to\infty$, and suppose that $\{Q_{N_i,\nu_i}\}_i$ is weakly convergent to a probability measure $Q$. Let $\mathcal{H}$ denote the set of viscosity solutions of the Euler equations starting at the initial datum $\xi_0$. Scaling-limit conjecture. The limit $Q$ is a probability measure supported on $\mathcal{H}$.

Earlier compactness arguments show that subsequential limits are probability distributions on weak solutions of the deterministic Euler equations, but uniqueness is not known. The conjecture asserts that allowing the dimension parameter $N_i$ to vary together with the vanishing viscosity parameter $\nu_i$ selects viscosity solutions, a stronger conclusion than the general superposition-solution result.