in such a way that $D$ contains unbounded components with positive probability.

To do this, we let $\tilde{C}$ be a Poisson Boolean model in $\hyp$ with intensity $\tilde{\lambda}$. Let $B$ be a bounded connected set in $\hyp$. Choose $\tilde{\lambda}$ so big that $\hyp$, we have $\E[L(B\cap \partial \tilde{C})]<\E[\mu(B\cap \tilde{C})]$. By Lemma 5.2 in [@tykesson], $\tilde{C}$ contains unbounded components with probability $1$. Let $\tilde{C}^D=\tilde{C}^D(y_1,y_2,r,n,\lambda)$ be the union of all balls in $\tilde{C}$ that are completely covered by $D$.

Suppose $E$ is some bounded connected set in $\hyp\times{\mathbb R}$. It is clear that $$\label{grans1}\lim_{r\rightarrow \infty}\P[E\subset
  A_1(r)]=1,$$ and that for fixed $r$, $$\label{grans2}\lim_{n\rightarrow \infty}\P[E\subset
  A_2(r,n)]=1,$$ and that for fixed $r$ and $n$, $$\label{grans3}\lim_{\lambda\uparrow\lambda_0}\P[E\subset
  A_3(r,n,\lambda)]=1.$$

Put $\delta:=\E[\mu(B\cap \tilde{C})]-\E[L(B\cap \tilde{C})]$. By (\[grans1\]), (\[grans2\]) and (\[grans3\]) we get that we can find first $r_1$ big enough, and then $n_1$ big enough, and finally $\lambda_1$ close enough to $\lambda_*$ so that $\E[\mu(B\cap
\tilde{C})]-\E[\mu(B\cap \tilde{C}^D)]<\delta/2$ and $\E[L(B\cap
\partial \tilde{C}^D)]-\E[L(B\cap \partial \tilde{C})]<\delta/2.$ With these choices of parameters, $\E[\mu(B\cap \tilde{C}^D)]>\E[L(B\cap
\tilde{C}^D)]$, so by Lemma 5.2 in [@tykesson], we get that $\tilde{C}^D$ contains unbounded components with positive probability. Since $\tilde{C}^D\subset D$, this implies that $D$ contains unbounded components with positive probability. Since the event that $D$ contains unbounded components is Isom($\hyp$)-invariant 