Let me first recall what is the Bressan conjecture. Take a $BV\cap L^\infty$ vector field $X$ on some open subset of $\mathbb R^n$ such that there exists an $L^\infty$ function $\alpha\ge 1$ so that $ \text{div}(\alpha X)=0 $ (such a vector field is said to be nearly incompressible). Let $\Sigma$ be a $C^1$ hypersurface non-characteristic with respect to $X$. Then the weak $L^\infty$ solutions of the equation $$ Xu=0, \quad u_{\vert \Sigma}=0, $$ are vanishing near $\Sigma$. In short that conjecture claims that the weak solutions of the Cauchy problem for a nearly incompressible $BV$ vector field are unique. The original version of the conjecture was dealing with compactness of the flows of approximations of a nearly incompressible vector field but anyhow this was proven equivalent to the previous uniqueness statement.
I heard a couple of years ago that the conjecture had been proven, but I have not seen any published article appearing in a Journal. So my question is "What is the status of the Bressan conjecture?"
