Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$.

So $G\in\mathcal{G}$ can be written $G(x)=\sum_{k=1}^\infty p_k x^k$ where $p_1, p_2, p_3, \dots$ are non-negative and sum to 1. Such a $G$ is a bijection from $[0,1]$ to $[0,1]$ whose derivatives exist and are non-negative everywhere on $[0,1)$.

Suppose $G\in\mathcal{G}$, and define $H(x)=1-G^{-1}(1-x)$. Is it necessarily the case that $H\in\mathcal{G}$?

(The motivation is closely related to that described in the question Involutions on $[0,1]$ given by power series (related to probability generating functions), which concerned the case where $H\equiv G$.)

EDIT: Thanks to Brendan McKay, who gives in the comments below the example $G(x)=\sum_{k\geq 2} \frac{x^k}{k(k-1)}$, for which one can show that $H''(0)$ diverges at $0$. I wonder if there are any natural general conditions on $G$ under which it does hold that $H$ must also be a probability generating function.