Uniqueness conjecture for the six-permutation quasirandom-forcing combination

Let $\rho^*$ be the quasirandom-forcing linear combination of six permutations

$$\rho^*:={\small \text{\tt{123}}}+{\small \text{\tt{321}}}+{\small \text{\tt{2143}}}+{\small \text{\tt{3412}}}+\frac{1}{2}\left({\small \text{\tt{2413}}}+{\small \text{\tt{3142}}}\right).$$

A linear combination is quasirandom-forcing if it has the quasirandomness-forcing property studied in the paper. Uniqueness conjecture. If $\rho$ is a quasirandom-forcing linear combination of six permutations, then

$$\rho=c\cdot\rho^*$$

for some $c\in\mathbb{R}$. The paper exhibits $\rho^*$ and proves that no positive linear combination of five or fewer permutations is quasirandom-forcing, but does not establish uniqueness or address the conjecture for arbitrary six-term combinations beyond the stated question for future study.