Pathwise uniqueness conjecture for Lagrangian trajectories of three-dimensional Leray solutions

Let $u_0\in L^2_\sigma$ be divergence-free initial data, let $u$ be an associated Leray solution of the three-dimensional Navier–Stokes equations, and consider the stochastic differential equation defining its Lagrangian trajectories from an initial point $x\in\mathbb{R}^3$. Pathwise uniqueness conjecture. For every $x\in\mathbb{R}^3$, pathwise uniqueness holds for this stochastic differential equation. This would extend almost-everywhere uniqueness to every initial point for arbitrary Leray solutions, a question left open by the theorem discussed in the paper; the source provides no resolution of the conjecture.