Deligne's period conjecture for motives of odd weight

Let $M$ be a $\mathbb{Q}$-motive with coefficients in a number field $F$ and odd weight. Let $H_B^+(M)$ be the $+1$-eigenspace of complex conjugation, let $\alpha_M^+$ be the comparison map from $H_B^+(M)\otimes\mathbb{C}$ to $H_{\operatorname{dR}}^+(M)\otimes\mathbb{C}$, and let $c^+(M,s_0)$ denote the Deligne period of $M$ at $s_0$. If $s_0\in\mathbb{Z}$ is critical for $M$, meaning that neither $L_\infty(M,s)$ nor $L_\infty(M^*(1),s)$ has a pole at $s_0$, then Deligne's period conjecture. The map $\alpha_M^+$ is an isomorphism and there exists $\mathcal{L}(M,s_0)\in F$ such that

$$L(M,s_0)=c^+(M,s_0)\cdot\mathcal{L}(M,s_0).$$

The conjecture predicts that Deligne periods give the irrational parts of critical motivic $L$-values. The paper applies this prediction to motivic pieces of superelliptic curves and numerically verifies instances, but the general statement remains open.