Renormalization conjecture for bounded nearly incompressible BV vector fields

Let $b$ be a bounded, nearly incompressible vector field in $\operatorname{BV}_{\rm loc}(\mathbb{R} \times \mathbb{R}^{d})$. A bounded, nearly incompressible vector field $b \in \operatorname{BV}_{\rm loc}(\mathbb{R} \times \mathbb{R}^{d})$ has the renormalization property in the sense that, for every $\rho$-weak solution $u \in L^{\infty}(I \times \mathbb{R}^{2})$ of the transport equation and every $\beta \in C^1(\mathbb{R})$, the function $\beta(u)$ is also a $\rho$-weak solution. Renormalization conjecture. Every bounded, nearly incompressible vector field $b \in \operatorname{BV}_{\rm loc}(\mathbb{R} \times \mathbb{R}^{d})$ has the renormalization property. This conjecture is presented as a statement that would imply a conjecture of A. Bressan; the supplied text gives no resolution status.