Asymptotic count of length-four continued-fraction strings with nontrivial symmetries

For a string $\mathbf{a}$ of positive integers, call a permutation-induced equality $P_{GK}(\sigma(\mathbf{a}))=P_{GK}(\mathbf{a})$ nontrivial when $\sigma(\mathbf{a})\ne\mathbf{a}$ and $\sigma(\mathbf{a})\ne\overleftarrow{\mathbf{a}}$. Let $f(N,4)$ be the number of unordered $4$-tuples of distinct positive integers at most $N$ having a permutation with a nontrivial symmetry, and let $\delta(N,4)=f(N,4)/\binom{N}{4}$.

Length-four symmetry-count conjecture.

$$\lim_{N\to\infty}\delta(N,4)=0$$

and

$$\lim_{N\to\infty}\frac{\log f(N,4)}{\log N}=1.$$

The second assertion is stronger than the first and predicts that the number of exceptional $4$-tuples has polynomial growth of exponent one, while the observed proportion tends to zero. The paper notes that its construction already gives the corresponding lower-bound exponent, but the upper bound remains open.