Failure of the Weak Sard property without Anomalous Dissipation

For every $\alpha \in (0,1)$ we construct an autonomous, divergence-free vector field $u\in C_c^{\alpha }({\mathbb{R}}^2,{\mathbb{R}}^2)$ that does not have the weak Sard property and, nonetheless, does not induce anomalous dissipation of $L^2({\mathbb{R}}^2)$ norm for solutions to the associated advection-diffusion equation. This disproves a conjecture proposed by Bagnara, Boutros, De Lellis and Mayboroda in \cite{BaBoDeMa26}.