Failure of the Weak Sard property without Anomalous Dissipation

For every $\alpha\in(0,1)$ we construct an autonomous, divergence-free vector field $u\in C^\alpha_c(\R^2,\R^2)$ that does not have the weak Sard property and, nonetheless, does not induce anomalous dissipation of $L^2(\R^2)$ norm for solutions to the associated advection-diffusion equation.

This disproves a conjecture proposed by Bagnara, Boutros, De Lellis and Mayboroda in \cite{BaBoDeMa26}.