The symmetry-group conjecture for Turk’s head knots

Let $p,q$ be positive integers with $\gcd(p,q)=1$, so that $Th(p,q)$ is a Turk’s head knot, and let $D_{2q}$ denote the dihedral group of order $2q$. Symmetry-group conjecture.

$$\operatorname{Sym}(S^3,Th(p,q))\cong D_{2q}.$$

The conjecture is motivated by computational experiments and is not resolved in the paper.