Lower asymptotic bound for primes in the alternating-sum sequence

Let $(A_n)$ be defined by $A_n=\sum_{i=1}^{2n}(-1)^ip_i$, and let $r_k=A_m$ be the $k$th prime occurring in this sequence. Alternating-sum prime lower-bound conjecture. For every $k\ge15234$,

$$r_k>\frac{m\sqrt{2m}\log m}{\sqrt{k\log k}}.$$

The claim is proposed from numerical data in the paper; no proof or resolution is supplied.