The shift-additivity conjecture for periodic binary sequences

Let $n$ be a positive integer, let $c\geq \left\lceil\frac{n}{2}\right\rceil$, and let $d=n-c$. Let $\mathcal{B}(n,c,d)$ denote the specified class of binary sequences, let $R^b(\mathbf{s}_n)$ denote the $b$-fold shift of $\mathbf{s}_n$, and let $\operatorname{add}(\mathbf{s}_n)$ denote its additivity parameter. Suppose that $0<b<n$ and that, for some $d_2$, $R^b(\mathbf{s}_n)\in\mathcal{B}(n,c,d_2)$ with $\operatorname{add}(R^b(\mathbf{s}_n))>\operatorname{add}(\mathbf{s}_n)$. Shift-additivity conjecture. Then $b=d_2$. The conjecture would extend the statements of the cited proposition and corollary from $c\geq\left\lceil\frac{2n-1}{3}\right\rceil$ to $c\geq\left\lceil\frac{n}{2}\right\rceil$, allowing the same conclusions about the unique relevant shift and the nonlinear complexity of the associated periodic sequence; the source reports experimental evidence but says that the proposed technique does not prove it.