Conjectured linear complexity of generalized cyclotomic binary sequences

Let $p$ be an odd prime, let $n\geq 1$, and let $s^\infty$ be the generalized cyclotomic binary sequence of period $p^n$ defined by the characteristic set $\mathcal{C}_1^{(p^n)}$ in the source. Let $D^{(p)}_0$ be the corresponding generalized cyclotomic class, and define

$$\delta(t)=\begin{cases}1,&\text{if }t\text{ is even},\\0,&\text{if }t\text{ is odd}. \end{cases}$$

Linear-complexity conjecture. If $2^{p-1}\not\equiv 1\pmod {p^2}$, then the linear complexity $L$ of $s^\infty$ is

$$L=\begin{cases}p^n-\dfrac{p-1}{2}-\delta\left(\dfrac{p^n+1}{2}\right),&\text{if }2\in D^{(p)}_0,\\p^n-\delta\left(\dfrac{p^n+1}{2}\right),&\text{if }2\notin D^{(p)}_0. \end{cases}$$

The claim predicts the linear complexity for all permitted primes and periods, extending the case $n=2$ discussed in the cited work; its resolution is not established by the supplied source context.