Infinite-order resolution ODE convergence conjecture

Let $g(z,s)$ be a discrete-time update, and let $f_i(Z)$ be the coefficient functions defined recursively by the paper's construction. Define

$$f^{(\infty)}(Z,s):=\sum_{i=0}^{\infty}f_i(Z)s^i.$$

For any $z$ and $z^+=g(z,s)$, let $Z(s)$ denote the solution at time $t=s$ of

$$\frac{dZ}{dt}=f^{(\infty)}(Z,s),\qquad Z(0)=z.$$

Infinite-order resolution ODE conjecture. Under certain regularity conditions on $g(z,s)$ and $s$, including, for example, that $g(z,s)$ is infinitely differentiable and that $s$ is sufficiently small, the series defining $f^{(\infty)}(Z,s)$ converges for every $Z$, and the resulting ODE satisfies

$$Z(s)=z^+.$$

This conjecture asserts that the infinite-order resolution ODE exactly reproduces one step of the discrete-time algorithm. The finite-order construction preceding it provides approximate resolution ODEs, but the supplied text does not specify precise sufficient regularity conditions or establish convergence and exactness in the infinite-order case.