The random-interval sequence k-width scaling conjecture

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

$$\lim_{n\rightarrow\infty}\frac{E[k\operatorname{-wd}(R_n)]}{n}=c_k,$$

and moreover

$$c_k=\frac{1}{k+1}.$$

The conjecture predicts linear scaling for the $k$-width of sequences of random intervals when $k\geq 2$, contrasting with the known behavior for $k=1$. The constants are motivated by computational experiments, while the asserted limit and formula remain open.