Direct entry indices preserve every finite acceptance-dependent history statistic

For all $\htmlData{mx=1}{n, d \in \htmlData{mx=0}{\mathbb{N}}}$, $\htmlData{mx=2}{E : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Type}} \cdot }}$, $\htmlData{mx=3}{A, q \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Key}}\,\htmlData{mx=1}{d}}}$, $\htmlData{mx=1}{w \in \htmlData{mx=0}{\mathbb{R}}}$, $\htmlData{mx=7}{k : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Key}}\,\htmlData{mx=4}{d}}}$, $\htmlData{mx=5}{v : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{E}}$, $\htmlData{mx=6}{\mathit{accept} : \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{Row}}\,\htmlData{mx=1}{d}\,\htmlData{mx=2}{E}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\mathbb{R}}}$, $\htmlData{mx=1}{m \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=12}{f : (\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{m}} \mathrel{\htmlData{mx=9}{\to}} \htmlData{mx=8}{\htmlData{mx=6}{\htmlData{mx=3}{\operatorname{Row}}\,\htmlData{mx=4}{d}\,\htmlData{mx=5}{E}} \times \htmlData{mx=7}{\operatorname{Bool}}}) \mathrel{\htmlData{mx=11}{\to}} \htmlData{mx=10}{\mathbb{R}}}$, if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{NormedAddCommGroup}}\,\htmlData{mx=1}{E}}$ and $\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{InnerProductSpace}}\,\htmlData{mx=1}{\mathbb{R}}\,\htmlData{mx=2}{E}}$), then