Lions' small-time global null controllability conjecture for the Navier–Stokes equation

Let $\Omega$ be the rectangular domain described above, and let $L^2_{\operatorname{div}}(\Omega)$ denote the space of square-integrable divergence-free vector fields tangent to $\Gamma_\pm$. Fix $T>0$ and $u_*\in L^2_{\operatorname{div}}(\Omega)$. Lions' controllability conjecture. There exists a weak Leray solution to the Navier–Stokes equation with zero distributed force and the prescribed Dirichlet boundary conditions such that

$$u(T,\cdot)=0.$$

This is the small-time global null controllability question attributed to Jacques-Louis Lions: every admissible initial state should be steerable to rest in any positive time without an external force in the formulation above. The supplied text does not indicate whether the conjecture has been resolved.