Conjecture on the common complexity of matrix transformations

Let $K=I\circ J$ be the transformation on matrices considered in the paper. Denote by $(G)$, $(S)$, and $(C)$ the patterns of general, symmetric, and cyclic matrices, respectively, and let $q$ be the parameter in the transformation.

Common-complexity conjecture. The complexity of $K$ for patterns $(G)$, $(S)$, and $(C)$ is the same. Its common value is the inverse of the modulus of the smaller root of

$$x^{2}-(q^{2}-4q+2)x+1=0.$$

The conjecture summarizes the agreement between the numerical and analytical approaches in the paper, which found the complexities for the three matrix patterns to be extremely close. The supplied text gives no resolution, so the conjecture remains open.