Omega sequences have linear complexity near half the sequence length

Let ${\cal O}$ denote the omega sequence, and let $L({\cal O},N)$ be its $N$th linear complexity.

Omega linear-complexity conjecture.

$$L({\cal O},N)=\frac{N}{2}+O(1).$$

This conjecture is based on numerical data; the paper gives no proof or further evidence of resolution.