The minimum size conjecture for quasirandom-forcing sets of permutations

A set of permutations is quasirandom-forcing if forcing the corresponding permutation densities determines the uniform measure on $[0,1]^2$. For permutations of arbitrary, possibly different, sizes, the smallest quasirandom-forcing set is known to have cardinality between four and six.

Minimum size conjecture for permutations. Every quasirandom-forcing set of permutations has cardinality at least six.

This conjecture asserts that the known upper bound of six is optimal. It strengthens the cited conjecture that no linear combination of six permutations forces quasirandomness.