The fitted asymptotic formula conjecture for peri-Catalan numbers

For a generator count $s$, let $P^s_n$ be the number of reduced quasigroup words of length $n$ on $s$ generators and let $C_n$ be the $n$-th Catalan number. Consider the normalized limit

$$L_s=\lim_{n\to\infty}\frac{\log P^s_n}{\log C_n+n\log 3s-\log3}.$$

Fitted asymptotic formula conjecture. For all generator counts $s$, the limit exists and approximately satisfies

$$L_s\simeq 1-\frac{0.019\dots}{s-\log\left(\frac{1+\sqrt{5}}{2}\right)}.$$

This formula is inferred from numerical data and a rational-function fit; it is intended to subsume the preceding unlabeled conjectures about the existence and dependence on $s$ of the normalized limits.