Generic non-precompactness of flow maps for divergence-free Sobolev velocities

Fix $p\in(1,\infty)$ and let

$$\mathcal{U}_p=\{u\in L^\infty([0,+\infty);W^{1,p}(\mathbb{T}^d;\mathbb{R}^d))\mid \nabla\cdot u=0\}.$$

For $u\in\mathcal{U}_p$, let $\{\Phi_t\}_{t\geq 0}$ be its associated flow maps.

Generic flow-map non-precompactness conjecture. For a residual subset of $\mathcal{U}_p$, the associated flow maps $\{\Phi_t\}_{t\geq 0}$ are not precompact in $L^1$.

The conjecture asks whether divergence-free velocity fields generically fail to generate precompact trajectories, in the Baire-generic sense. The source does not provide evidence of a resolution.