Legendre sequence has near-maximal 2-adic complexity profile

Let ${\cal L}_p$ be the Legendre sequence modulo a prime $p$, and let $C({\cal L}_p,N)$ denote its $N$th $2$-adic complexity, defined as the binary logarithm of

$$\min\{\max\{|f|,|g|\}: f,g\in \mathbb{Z},\ g\text{ odd, }gS(2)\equiv f\bmod 2^N\},$$

where

$$S(2)=\sum_{n=0}^{N-1}s_n2^n.$$

Legendre 2-adic-complexity conjecture.

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

The conjecture is motivated by numerical data obtained with a rational approximation algorithm. The expected value of the aperiodic $2$-adic complexity is stated to be an open question, and no resolution of this specific conjecture is given.