Conjectural Sato–Tate group formula for even-genus trinomial hyperelliptic curves

Let $C_1$ be the hyperelliptic curve

$$y^2=x^{2g+2}+c,$$

where $g=2k$ is its genus and $c$ is the constant appearing in the defining equation. Let $\operatorname{ST}^0(C_1)$ denote the identity component of the Sato–Tate group of the Jacobian of $C_1$. Even-genus Sato–Tate group conjecture. One has

$$\operatorname{ST}^0(C_1)=\underbrace{\operatorname{U}(1)_2\times\operatorname{U}(1)_2\times\cdots\times\operatorname{U}(1)_2}_{k\text{ times}}.$$

The formula is motivated by computations using the paper's algorithm for genera $2$, $4$, $6$, and $8$; its validity for all even genera remains conjectural in the supplied text.