Non-uniqueness conjecture for 2D Euler solutions with $L^p$ vorticity

Let $2<p<\infty$. Consider the two-dimensional Euler equation with initial velocity $v^\circ\in L^2$ and initial vorticity $\omega^\circ\in L^1\cap L^p$. A weak solution has velocity $v\in C_t L^2$ and vorticity $\omega\in L_t^\infty(L^1\cap L^p)$. Non-uniqueness conjecture. For every $2<p<\infty$, there exists an initial velocity $v^\circ\in L^2$ with $\omega^\circ\in L^1\cap L^p$ such that more than one weak solution $v\in C_t L^2$ with $\omega\in L_t^\infty(L^1\cap L^p)$ exists for the Euler equation. Global existence for the two-dimensional Euler equation has been proved in other vorticity classes, but uniqueness in this class is not expected from the available regularity theory and remains open.