Noise sensitivity of Poisson Boolean models with bounded simply connected shapes

Let $S\subset\mathbb{R}^2$ be a bounded simply connected shape. Given a Poisson point process $\eta$, form the occupied set by placing a translate $x+S$ at every point $x\in\eta$.

Shape-noise-sensitivity conjecture. If $S$ is bounded and simply connected, then the Poisson Boolean model for $S$ is noise sensitive at criticality.

This extends the question from discs to deterministic shapes; the conjecture is presented as an open problem, while the corresponding model with random shapes and the higher-dimensional case are described as further questions.