Restricted Prime Number Theorem for shifted-index alternating sums

Let $p_i$ be the $i$th prime. For a fixed positive integer $k$, define

$$A_n^{(k)}=\sum_{i=1}^{2n+1}(-1)^{i-1}p_{i+k},\qquad n=1,2,\ldots.$$

Shifted-index restricted-prime conjecture. For every fixed positive integer $k$, the sequence $(A_n^{(k)})$ satisfies the Restricted Prime Number Theorem. The conjecture is presented as a generalization motivated by heuristic arguments, computation, and the main alternating-sum conjecture; no resolution is stated.