Noise sensitivity of the Poisson Boolean model with random radii

Let $R>0$ and let $\mu_R$ be a probability distribution on $(0,R)$. In the corresponding planar Poisson Boolean model, independently assign a radius with distribution $\mu_R$ to each point of a Poisson point process.

Random-radius noise-sensitivity conjecture. For every $R>0$ and $\mu_R$, the Poisson Boolean model with random radii chosen according to $\mu_R$ is noise sensitive at criticality.

This is a natural extension of the fixed-radius model; the proposed method would require an RSW theorem for the occupied space, and the conjecture remains open.