Remizov's dense second-order approximation-subspace conjecture

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 $L$. Fix $t_0\geq0$, and let $D(t_0)$ denote the intersection, over $t\in[0,t_0]$, of the domains of $G'(t)$, $G”(t)$, $G”'(t)$, $G””(t)$, $G'(t)G”(t)$, $G'(t)^2G”(t)$, and $G”(t)^2$. Assume that $D(t_0)$ is dense in $\mathcal{F}$ and that each of these operators depends continuously on $t$ on every vector in $D(t_0)$. For $\zeta(n)=1/n^2$, define

$$A^{[0,t_0)}_{\zeta}=\left\{f\in\mathcal{F}:\sup_{t\in[0,t_0)}\left\|\left(G\left(\frac{t}{n}\right)\right)^nf-e^{tL}f\right\|=O(\zeta(n))\text{ as }n\to\infty\right\}.$$

Remizov's dense second-order approximation-subspace conjecture. The function $G$ has a dense approximation subspace $A^{[0,t_0)}_{\zeta}$ of order $\zeta(n)=1/n^2$, and $A^{[0,t_0)}_{\zeta}\subseteq D(t_0)$. This reformulation asserts that sufficiently regular data form a dense class with uniform second-order Chernoff convergence on $[0,t_0)$; the source gives no resolution.