Asymptotic Briggs inequalities for regular partitions and overpartitions

Let $p_k(n)$ and $\overline{p}_k(n)$ denote the $k$-regular partition and $k$-regular overpartition functions, respectively, and let the inequalities referred to in the source as (p_kmain) and (opk-main) be the corresponding Briggs inequalities. The conjecture. For every positive integer $k\geq 2$, there exists $N(k)$ such that both inequalities hold for all $n\geq N(k)$. The paper proves the claim in certain ranges of $k$ and discusses computational evidence for $2\leq k\leq 100$, but the supplied text does not establish the conjecture for every $k$.