The stronger asymptotic error-bound conjecture for associated Stirling numbers

Let $S_r(p,q)$ denote the associated Stirling numbers and let $F_r(p,q)$ denote the approximation defined earlier in the paper. Fix a positive integer $r$.

Stronger asymptotic error-bound conjecture. As $p\to\infty$,

$$S_r(p,q)=F_r(p,q)(1+O(p^{-1})).$$

This proposes a stronger error estimate for the asymptotic approximation. The supplied text gives no resolution or additional hypotheses on $q$, so the exact intended asymptotic regime should be checked in the paper.