Asymptotic constant for the discrepancy of the lex-least de Bruijn sequence

Let ${\mathcal L}_n$ be the lexicographically least binary order-$n$ de Bruijn sequence, formed by concatenating all Lyndon words of length dividing $n$ in lexicographic order. For a word $w$, define its discrepancy by

$$\operatorname{disc}(w)=\max_M\left|\sum_{k=0}^M(-1)^{w_k}\right|.$$

Discrepancy asymptotic constant conjecture. There is some constant $C$ such that

$$\lim_{n\rightarrow\infty}\frac{n\operatorname{disc}({\mathcal L}_n)}{2^n\log n}=C.$$

The preceding theorem establishes the order of magnitude ${\operatorname{disc}({\mathcal L}_n)}=\Theta(2^n\log n/n)$; the conjecture asserts that the normalized discrepancy has a limiting constant, strengthening that asymptotic estimate.