The spectral characterization conjecture for convex subsets of the plane

Let $\mathcal C$ be a proper closed convex subset of $\mathbb R^2$ with positive Lebesgue measure. Assume that $\operatorname{Op}^w(\mathbf 1_{\mathcal C})$ is a bounded self-adjoint operator on $L^2(\mathbb R)$ and that its spectrum is contained in $[0,1]$. Spectral characterization conjecture for convex subsets of the plane. Then $\mathcal C$ is, up to an affine symplectic map, the strip

$$[0,1]\times\mathbb R.$$

The boundedness assumption would be unnecessary if the bounded-spectrum conjecture for convex bodies were proved. The claim seeks to characterize the exceptional convex domains whose Weyl-quantized indicators have spectrum contained in the classical interval $[0,1]$; it remains open.