The random-interval set k-width scaling conjecture

Let $k\geq 2$, and let $W_n$ be a set of $n$ random intervals. Write $E[k\operatorname{-wd}(W_n)]$ for the expected $k$-width of $W_n$. Random-interval set scaling conjecture. There exists a positive constant $d_k>0$ such that

$$\lim_{n\rightarrow\infty}\frac{E[k\operatorname{-wd}(W_n)]}{n}=d_k.$$

The experiments suggest that $d_2=c_2=\frac{1}{3}$ and, for $k=3,4$, $d_k=c_k=\frac{1}{k+1}$. The conjecture extends this observed pattern to all $k\geq 2$:

$$d_k=c_k=\frac{1}{k+1}\quad\text{for all }k\geq 2.$$

The claim concerns the asymptotic $k$-width of sets of random intervals; the evidence given is experimental, and the general result remains open.