Higher-order asymptotic conjecture for parabolic double cosets

Let $p_n$ be the number of parabolic double cosets in $S_n$, and let

$$K=\frac{e^{-(\log 2)^2/2}}{4(\log 2)^2}.$$

Higher-order asymptotic conjecture. There exists a constant $c>0$ such that

$$\frac{p_n(\log 2)^{2n}}{n!}=K-\frac{c}{n}+O\left(\frac{1}{n^2}\right).$$

This is motivated by numerical data and is presented as a further-direction conjecture; no resolution is given in the supplied text.