Non-uniqueness conjecture for the Sobolev divergence chain rule

Let $p,\tilde p \in [1, \infty)$ and assume that

$$\frac{1}{p}+\frac{1}{\tilde p}>1+\frac{1}{d}.$$

Let $\beta:\mathbb{R}\to\mathbb{R}$ and let $T\in\mathcal{D}'(\mathbb{T}^d)$ satisfy the suitable assumptions specified in the source. An incompressible vector field is a vector field $u$ with vanishing distributional divergence. The conjecture asserts that there exist a density $\rho\in L^p(\mathbb{T}^d)$ and an incompressible vector field $u\in L^{p'}(\mathbb{T}^d)\cap W^{1,\tilde p}(\mathbb{T}^d)$ such that the equations referenced in the source as

$$and$$

hold simultaneously. In certain cases, $\rho$ can moreover be chosen strictly positive.

Non-uniqueness conjecture. Under the stated integrability condition and suitable assumptions on $\beta$ and $T$, such a density $\rho$ and incompressible Sobolev vector field $u$ exist, with the divergence equation and chain-rule defect equation holding simultaneously; in certain cases the density may be chosen strictly positive.

The conjecture is motivated by the analogy between the transport equation and the divergence chain-rule problem, together with convex-integration non-uniqueness results for transport by Sobolev vector fields. The source presents the assumptions on $\beta$ and $T$ only by reference to later points, and does not provide a resolution of this conjecture.