The twisting Sato-Tate group of the curve y² = x⁸ - 14x⁴ + 1

We determine the twisting Sato-Tate group of the genus $3$ hyperelliptic curve $y^2 = x^{8} - 14x^4 + 1$ and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of $y^2 = x^{8} - 14x^4 + 1$. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all $\mathbb Q$-twists of the curve $y^2 = x^{8} - 14x^4 + 1$.