Generic boundedness conjecture for inviscid limits in the critical Besov class

Let $u_0\in L^2$ be initial data, and consider sequences of Leray–Hopf weak solutions of the Navier–Stokes equations whose inviscid limits are denoted by $u$. The critical Besov class is $L^3_t B^{1/3}_{3,\infty}$. Generic boundedness conjecture. For generic initial conditions $u_0\in L^2$, inviscid limits of sequences of Leray–Hopf weak solutions of the Navier–Stokes equations remain uniformly bounded in

$$L^3_t B^{1/3}_{3,\infty}.$$

This would identify critical Besov regularity as a generic uniform bound for inviscid limits and would connect the Kolmogorov scaling laws with compactness and regularity properties of vanishing-viscosity solutions. The conjecture is presented as open in the source.