Remizov's second-order asymptotic conjecture for Chernoff approximations

Let $(e^{tL})_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $\mathcal{F}$ with generator $(L,\operatorname{dom}(L))$, and let $G$ be a Chernoff function for $L$. Assume that $t_0\geq 0$ and $f\in\mathcal{F}$ are given, and that, for every $t\in[0,t_0]$, $f$ belongs to the domains of $G'(t)$, $G”(t)$, $G”'(t)$, $G””(t)$, $G'(t)G”(t)$, $G'(t)^2G”(t)$, and $G”(t)^2$, with each corresponding operator continuous in $t$. Remizov's second-order asymptotic conjecture. There exists $C\geq0$ such that, for every $t\in[0,t_0)$ and every $n\in\mathbb{N}$,

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

This refines the expected first-order Chernoff convergence by identifying a correction term and an $O(n^{-2})$ remainder; the source gives no resolution.