The bi-canonical degree comparison conjecture

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring that has a canonical ideal $\mathcal{C}$. The canonical degree and bi-canonical degree are denoted by $\operatorname{cdeg}(R)$ and $\operatorname{bideg}(R)$, respectively. Comparison conjecture. In general,

$$\operatorname{bideg}(R) \leq \operatorname{cdeg}(R).$$

The conjecture compares two numerical measures of the deviation of a Cohen–Macaulay ring from being Gorenstein; its resolution is not established in the supplied text.