 Solution

The union area is a bounded measurable function of the centres, with $0\le X\le n\pi$. Fix all centres except one, and let $B$ be the union of the fixed disks. Replacing the remaining disk $D$ changes the area to $|B|+|D\setminus B|$. The second term always lies in $[0,\pi ]$, so the area changes by at most $\pi$, not merely by the weaker bound $2\pi$.

Thus $X/\pi$ has coordinate differences at most one for the $n$ independent centres. The preceding martingale bound givesOne may take $c=1/(2{\pi }^2)$, independently of all centre distributions. Boundedness of the area ensures the conditional expectations exist even when the centre locations have no finite moments.