The asymptotic irrelevance conjecture for quasigroup identities

Let $P^s_n$ denote the $n$-th $s$-peri-Catalan number, counting reduced quasigroup words of length $n$ on $s$ generators. “In the large” refers informally to large values of $s$ for almost all values of $n$.

Asymptotic irrelevance conjecture. In the large, cancellation resulting from the quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers $P^s_n$.

The conjecture is motivated by the expectation that exact matching of variable assignments required for cancellation becomes relatively unlikely when many generators are available; its asymptotic consequences are investigated numerically in the paper.