Conjecture on the gap between Poisson Stein-factor constants

Let $\lambda>0$ and let $k$ be an integer with $k>\lambda$. For the solution of the Poisson Stein equation with threshold test function $h={\bf 1}_{[k,\infty)}$, let ${\mathds{C}}_{1-}(\lambda,k)$ and ${\mathds{C}}_{1+}(\lambda,k)$ denote the lower- and upper-side first-difference constants, respectively:

$${\mathds{C}}_{1-}(\lambda,k):=\frac{F(k-1)}{k\pi_k}\left(1-\frac{F(k-2)}{F(k-1)}\cdot\frac{\lambda}{k-1}\right),$$ $${\mathds{C}}_{1+}(\lambda,k):=\frac{F(k-1)}{k\pi_k}\left(1-\frac{\overline{F}(k+1)}{\overline{F}(k)}\cdot\frac{k}{\lambda}\right).$$

Stein-factor gap conjecture. For all $k>\lambda$,

$${\mathds{C}}_{1-}(\lambda,k)>{\mathds{C}}_{1+}(\lambda,k),$$

and the gap increases exponentially as a function of $k-\lambda$.

The conjecture is motivated by the birth–death interpretation of the two constants and would quantify asymmetric behavior of Poisson Stein factors above the mean. The source provides no resolution status or proof.