The non-negative coefficient conjecture for the associated Stirling series

Let $B_r(x)$ denote the function used in the associated Stirling-number asymptotics, and let $r$ be an integer with $r\geq 1$. Consider the power series about $0$ of

$$e^{-x/(r+1)}B_r(x).$$

Non-negative coefficient conjecture. For each integer $r\geq 1$, this power series has all non-negative coefficients.

If true, this would simplify the proof of the asymptotic expansions for associated Stirling numbers, in particular the proof that $|J|\to 0$. The supplied text gives no resolution of the conjecture.