What is the number of bijections $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ such that, for all distinct $i, j \in\{1, \ldots, 9\}$, whenever the squares labelled $i$ and $j$ in the diagram below share an edge, the squares labelled $f(i)$ and $f(j)$ share an edge too?
$$\begin{array}{|l|l|l|}
\hline 1 & 2 & 3 \\
\hline 4 & 5 & 6 \\
\hline 7 & 8 & 9 \\
\hline
\end{array}$$
- $8$
- $9$
- $4$
- $24$
