The comparison conjecture for canonical and bi-canonical degree

Let $({\mathbf R},{\mathfrak m})$ be a one-dimensional Cohen–Macaulay local ring with a canonical ideal $\mathcal{C}$. If $(c)$ is a minimal reduction of $\mathcal{C}$, define

$$\operatorname{cdeg}({\mathbf R})=\lambda({\mathbf R}/(c))-\lambda({\mathbf R}/\mathcal{C}),\qquad \operatorname{bideg}({\mathbf R})=\lambda({\mathbf R}/\mathcal{C})-\lambda({\mathbf R}/\mathcal{C}^{**}).$$

Comparison conjecture. In general,

$$\operatorname{cdeg}({\mathbf R})\geq \operatorname{bideg}({\mathbf R}).$$

The conjecture proposes a general comparison between the canonical degree and the bi-canonical degree, two numerical invariants measuring the deviation of a Cohen–Macaulay ring from being Gorenstein. Its resolution is not supplied in the source.