The conjecture on polynomial-size join/union expressions for permutation cycles

Let $[n]=\{1,\ldots,n\}$, let $\mathfrak{S}_n$ be the set of permutations of $[n]$, and let $\mathfrak{C}_n$ be the set of $n$-cycles in $\mathfrak{S}_n$. A join/union expression is an expression built using the paper's join and union operations that represents a set of maps. Cycle-expression conjecture. For every polynomial $P$, there exists an integer $n$ such that $\mathfrak{C}_n$ does not have a join/union expression of size at most $P(n)$, even if $n$ is represented in unary. The preceding theorem gives a conditional obstruction: a polynomial-time construction would imply ${\sf P}={\sf NP}$. The source presents the unconditional size lower bound as an open conjecture.