Remizov's first-order convergence 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 $t\in[0,t_0]$, $f\in\operatorname{dom}(G'(t))\cap\operatorname{dom}(G”(t))$ and the functions $t\mapsto G'(t)f$ and $t\mapsto G”(t)f$ are continuous. Remizov's first-order convergence conjecture. There exists $C\geq 0$ such that, for every $t\in[0,t_0)$ and every $n\in\mathbb{N}$,

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

This conjecture concerns quantitative convergence in the Chernoff product formula; under the stated regularity assumptions, the expected convergence rate is first order, while the source provides no resolution.