Powerlessness conjecture for late changepoint detection below the square-root scale

Consider the preferential attachment model with changepoint time $\tau_n=n-cn^\gamma$ for $c>0$ and $\gamma\in(0,1)$, and let $D_v(n)$ denote the degree of vertex $v$ in the final network and $G_n$ the final network snapshot. A test is powerful if its power tends to one while its type-I error tends to zero.

Powerlessness conjecture. When $\gamma<\tfrac{1}{2}$, all tests based on the vertex degrees $\{D_v(n)\}_{v\in[n]}$ are powerless, and all tests based on $G_n$ are powerless.

The second assertion is stronger because it rules out using higher-level information in the edge structure of $G_n$. The question is formally open; the conjecture is motivated by the fact that degree-count mean shifts are of order $n^\gamma$, below their fluctuation scale when $\gamma<\tfrac{1}{2}$, and by asymptotic normality results suggesting that the corresponding distributions have vanishing total variation distance.