The minimum size conjecture for quasirandom-forcing sets of 4-point permutations

A set of permutations is quasirandom-forcing if forcing the corresponding permutation densities determines the uniform measure on $[0,1]^2$. The paper has shown that every such set of $4$-point permutations has at least five elements, and examples with eight elements are known.

Minimum size conjecture for 4-point permutations. Every quasirandom-forcing set of $4$-point permutations has cardinality at least eight.

The conjecture asserts that the known upper bound of eight is tight, so the minimum cardinality is exactly eight.