Conjecture on minimally knotted cycles in canonical book representations

Let $K_n$ be the complete graph on $n$ vertices, and let $\widetilde{K}_n$ denote its canonical book representation. A knotted cycle is a cycle whose spatial embedding is a non-trivial knot, and an embedding of $K_n$ may contain knotted cycles that are not Hamiltonian.

Minimal knotted-cycle conjecture. The canonical book representation $\widetilde{K}_n$ contains the fewest total number of knotted cycles possible in any embedding of $K_n$.

The conjecture is motivated by computed counts for $8\leq n\leq 11$ and by lower and upper bounds for the minimum number of knotted cycles in embeddings of $K_8$; the optimality of the canonical representation remains open.