Second-order Chernoff approximation error conjecture

Let $(e^{tL})_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $\mathcal{F}$ with generator $(L,D(L))$. Let $G$ be a Chernoff function for $L$ satisfying condition (N) or (N'). Fix $t_0\geq 0$, and suppose that, for all $t\in[0,t_0]$, $f$ belongs to the domain of $L^4$. Second-order Chernoff approximation conjecture. Under possibly additional technical assumptions, there exists $C\geq 0$ such that, for all $t\in[0,t_0)$ and $n\in\mathbb{N}$,

\left\\|\left(G\left(\frac{t}{n}\right)\right)^n f-e^{tL}f+\frac{t^2}{2n}e^{tL}\left(L^2-G”(0)\right)f\right\\|\leq\frac{C}{n^2}.

This is a proposed simplification of a previously cited conjecture concerning higher-order convergence of Chernoff approximations; the phrase “maybe under some other technical assumptions” indicates that the precise hypotheses needed for the estimate are not specified, so its resolution is unclear.