Bressan's mixing cost conjecture for incompressible flows

Let $H$ be the left half of the torus, and let $\Psi$ be the flow generated by an incompressible vector field $u$. The image of $H$ under the flow $\Psi$ after time $T$ is called $\delta$-mixed if it has the mixing property at scale $\delta$. Bressan's conjecture. If after time $T$ the image of $H$ under the flow $\Psi$ is $\delta$-mixed, then there exists a constant $C$ such that

$$\int_0^T \\|\nabla u(\cdot,t)\\|_{L^1}\,dt \geqslant \frac{|\ln \delta|}{C}.$$

This conjecture asserts a logarithmic lower bound on the accumulated spatial gradient needed to mix a set to scale $\delta$. It is a formulation of the mixing-cost conjecture attributed to Bressan and remains open to the authors' knowledge.