Classify commutativity of shuffle elements for almost all index pairs

Determine whether, for sufficiently large \(n\), the shuffle elements \(\mathscr{Y}_{n,i}\) and \(\mathscr{Y}_{n,j}\) are noncommutative for almost all distinct pairs \(1\leq i,j\leq \binom{n}{2}\), and prove or refute the proposed monotonicity that noncommutativity of \(\mathscr{Y}_{n,i}\) and \(\mathscr{Y}_{n,j}\) implies noncommutativity of \(\mathscr{Y}_{n+1,i}\) and \(\mathscr{Y}_{n+1,j}\).

Background

The paper disproves the previously proposed universal commutativity claim by proving that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ do not commute for every $n\geq4$. It also gives examples of both commuting and noncommuting pairs for small values of $n$. The authors then formulate a broader unresolved conjectural pattern concerning the prevalence and persistence of noncommutativity among pairs of shuffle elements as the rank increases.