The linear-complexity spectrum conjecture for de Bruijn sequences

Let $n\geq 3$, let $c$ be an integer satisfying

$$2^{n-1}+n\leq c\leq 2^n-1,$$

and suppose $c\neq 2^{n-1}+n+1$. A span $n$ de Bruijn sequence is a cyclic binary sequence in which every binary word of length $n$ occurs exactly once. The linear-complexity spectrum conjecture. For every such $c$, there exists a span $n$ de Bruijn sequence ${\cal S}$ with

$$c({\cal S})=c.$$

The preceding theorem establishes the bounds and the exclusion of $2^{n-1}+n+1$, and realizes the two endpoint values $2^{n-1}+n$ and $2^n-1$; the conjecture asks for all remaining admissible linear complexities.