Sharpness of the Yudovich regularity-loss estimate

Let $\omega_0\in H^1(\mathbb{T}^2)\cap L^\infty(\mathbb{T}^2)$ be mean-zero, and let the corresponding solution of the two-dimensional incompressible Euler equations be a Yudovich solution. The estimate in the preceding lemma gives control of the evolving $W^{1,p}$ norm only while $p$ is below the decreasing exponent determined by the lemma. Sharpness of the regularity-loss estimate. The bound in the lemma is sharp, in the sense that there exist Yudovich solutions which continuously lose regularity. This would show that the further degeneration allowed by the existing estimates is genuinely possible; the statement is posed as an open problem, and no construction establishing it is supplied here.