Bressan's mixing conjecture for Sobolev velocity fields

Let $b\in L^{\infty}([0,\infty),W^{1,1}(\mathbb{T}^d,\mathbb{R}^d))$ be divergence free, and let $\rho$ denote the mixing rate associated with an initial datum $u_0$. Bressan's mixing conjecture. There exist $c>0$ and $C>0$, depending only on $u_0$, such that

$$\rho(t)\ge C\exp\left\lbrace-ct\left\lVert\nabla b\right\rVert_{L^{\infty}_tL^1_x}\right\rbrace$$

for every $t\geq 0$. The conjecture asserts a universal exponential lower bound on mixing in the nonsmooth setting and is presented here as an open question.