Almost all continued-fraction strings have no nontrivial symmetries

Let $\mathbf{a}=(a_1,\dots,a_n)$ be a string of positive integers. A nontrivial symmetry of $\mathbf{a}$ is a permutation $\sigma$ of $1,\dots,n$ such that $\sigma(\mathbf{a})\ne\mathbf{a}$, $\sigma(\mathbf{a})\ne\overleftarrow{\mathbf{a}}$, and $P_{GK}(\sigma(\mathbf{a}))=P_{GK}(\mathbf{a})$. For $n\ge4$, let $\delta(N,n)$ denote the proportion of unordered $n$-tuples of distinct positive integers at most $N$ that have a permutation with a nontrivial symmetry.

Almost-symmetry-free strings conjecture. For any integer $n\ge4$,

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

The conjecture generalizes the observed behavior for strings of lengths $4$, $5$, and $6$, and would imply that nontrivial symmetries are rare among strings of every fixed length at least four. The paper presents numerical evidence but does not establish the limit in general.