Deligne's period conjecture for superelliptic curve motivic pieces

Let $C/\mathbb{Q}(\zeta_m)$ be a superelliptic curve $y^m=f(x)$ as in the paper, let $\tau$ be a character of $C_m$, and fix an embedding $\mathbb{Q}(\zeta_m)\hookrightarrow\mathbb{C}$. For each $\sigma\in\operatorname{Gal}(\mathbb{Q}(\tau)/\mathbb{Q})$, let $\Omega_{C^{\sigma(\tau)}}$ denote the corresponding period. Deligne's period conjecture for superelliptic pieces. There exists $\mathcal{L}(C^\tau)\in\mathbb{Q}(\zeta_m)$ such that

$$L(C^{\sigma(\tau)},1)=\Omega_{C^{\sigma(\tau)}}\cdot\sigma(\mathcal{L}(C^\tau)).$$

This is the specialization of Deligne's period conjecture that the paper investigates for superelliptic curves. The authors report numerical verification for previously uninvestigated motivic-piece $L$-functions, while the assertion itself remains conjectural.