The comparison conjecture for canonical and bi-canonical degrees

Let $R$ be a Cohen–Macaulay local ring with a canonical ideal, and let $\text{cdeg}(R)$ and $\text{bideg}(R)$ denote its canonical degree and bi-canonical degree, respectively.

Comparison conjecture. In general,

$$\text{cdeg}(R) \geq \text{bideg}(R).$$

The conjecture compares two numerical invariants of Cohen–Macaulay local rings. The surrounding results establish nonnegativity and vanishing criteria for the bi-canonical degree, as well as formulas in several classes of one-dimensional and monomial rings, but do not resolve the inequality in general.