Bressan's compactness conjecture for nearly incompressible flows

Let $b_k:[0,+\infty[\times\mathbb{R}^d\to\mathbb{R}^d$, $k\in\mathbb{N}$, be a sequence of smooth vector fields, and let $X_k$ denote their classical flows. Assume that

$$\|b_k\|_\infty+\|\nabla b_k\|_{L^1}$$

is uniformly bounded, and that the flows are uniformly nearly incompressible: for some constant $C>0$,

$$\frac{1}{C}\leq{\rm \det}(\nabla_xX_k(t,x))\leq C.$$

Bressan's compactness conjecture. Under these assumptions, the sequence $\{X_k\}$ is strongly precompact in $L^1_{\operatorname{loc}}([0,+\infty[\times\mathbb{R}^d)$.

This conjecture concerns compactness of flows under uniform bounds on the vector fields and their gradients and a uniform nondegeneracy condition on the Jacobians. The source notes that a positive answer to the renormalization conjecture would imply a positive answer to this compactness conjecture.