The increasing property for binomial and Poisson distributions

Let $M$ be a binomially or Poisson-distributed random variable with $G$-transform $G(M)$, and let $\Phi$ denote the standard Gaussian distribution function. The increasing property. The function

$$m\longmapsto\frac{P(M=m)}{\Phi\left(G(m+1)\right)-\Phi\left(G(m)\right)}$$

is increasing, while the function

$$m\longmapsto\frac{P(M=m)}{\Phi\left(G(m)\right)-\Phi\left(G(m-1)\right)}$$

is decreasing. The source describes this as a stronger conjecture that implies the intersection property for binomial and Poisson distributions, but gives no resolution of the conjecture.