Narayana distribution conjecture for path lengths of 231-avoiding permutations

Let $S_n(231)$ denote the set of $231$-avoiding permutations of $[n]$. For $\pi\in S_n(231)$, let $\operatorname{pathlen}(\pi)$ be the length of the path corresponding to $\pi$ under the construction above, and let $N(n,k)$ be a Narayana number.

Narayana distribution conjecture. The path lengths have the Narayana distribution:

$$\sum_{\pi \in S_n(231)} q^{\operatorname{pathlen}(\pi)} = q^{2n} \sum_{k \ge 0} N(n,k)q^{2k}.$$

Moreover,

$$\sum_{n \ge 0}\sum_{\pi \in S_n(231)}q^{\operatorname{pathlen}(\pi)} = \frac{1-q^2+q^4-\sqrt{1-2q^2-q^4-2q^6+q^8}}{2q^4}.$$

The conjecture identifies the length statistic arising from the bijective construction with the Narayana distribution and gives its bivariate generating function. The paper presents this as an observed pattern supported by computations; no proof or resolution is supplied here.