Perron's capacity of random sets

Given a sequence of random variables $\{X_k:k\ge 1\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $${\mathrm{\Omega }}_{\text{rand},\text{lin}}:=\{\frac{\pi X_k}{k}:k\ge 1\}$$ and $${\mathrm{\Omega }}_{\text{rand},\text{lac}}:=\{\frac{\pi X_k}{2^k}:k\ge 1\}\mathrm{.}$$ We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1<p<\mathrm{\infty }$.