Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf 1_C^{Weyl}u,u\rangle\le \Vert u\Vert_{L^2(\mathbb R^n)}^2,\tag F $$ or more explicitly, $ \iint_{\mathbb R^n\times\mathbb R^n}\mathbf 1_C(x,\xi)\mathscr H(u,u)(x,\xi) dx d\xi\le \Vert u\Vert_{L^2(\mathbb R^n)}^2, $ where $\mathscr H(u,u)$ is the Wigner function given by $$ \mathscr H(u,u)(x,\xi)=\int_{\mathbb R^n}e^{-2i\pi z\cdot \xi} u(x+\frac{z}{2})\bar u(x-\frac{z}{2}) dz. \tag W$$ Since $(W)$ is real-valued, the lhs of $(F)$ is also real. The property is easy for $C$ equal to a half-space, since, due to Segal identity on symplectic transformations of Weyl symbols, the operator with Weyl symbol $\mathbf 1_C$ is, in that case, unitarily equivalent to the multiplication by $H(x_1)$.

Inequality (F) seems to have received the name of Flandrin's conjecture.