The intersection property for binomial and Poisson distributions

Let $M$ be a binomially distributed or Poisson-distributed random variable, let $P_0$ be its distribution, and let $G(M)$ be its $G$-transform, defined by assigning the signed square root of twice the divergence from $P_0$ according as the value is below or above the mean. Let $Z$ be a standard Gaussian. The intersection property. The quantile transform between $G(M)$ and $Z$ is a step function, and the identity function intersects each step, namely

$$P(M<m)<P\left(Z\leq G(m)\right)<P(M\leq m)$$

for every integer $m$. This conjecture asserts a precise interlacing between the discrete distribution and the Gaussian distribution after the $G$-transform; the source presents it as supported by quantile plots, and no resolution is supplied.