Subcritical sharpness for multiscale Boolean percolation

We consider a multiscale Boolean percolation on $\mathbb R^d$ with radius distribution $\mu$ on $[1,+\infty)$, $d\ge 2$. The model is defined by superposing the original Boolean percolation model with radius distribution $\mu$ with a countable number of scaled independent copies. The $n$-th copy is a Boolean percolation with radius distribution $\mu|_{[1,\kappa]}$ rescaled by $\kappa^{n}$. We prove that under some regularity assumption on $\mu$, the subcritical phase of the multiscale model is sharp for $\kappa$ large enough. Moreover, we prove that the existence of an unbounded connected component depends only on the fractal part (and not of the balls with radius larger than $1$).