The comparison conjecture for canonical and bidual canonical degrees

Let $({\mathbf R},{\mathfrak m})$ be a Cohen--Macaulay local ring of dimension $1$ with canonical ideal $\mathcal{C}$. Let $(c)$ be a minimal reduction of $\mathcal{C}$, so that $\mathcal{C}^{**}=(c):((c):\mathcal{C})$.

Comparison conjecture. The inequality

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

holds. Equivalently,

$$\lambda(\mathcal{C}/(c))\geq\lambda(\mathcal{C}^{**}/\mathcal{C}),$$

or

$$\operatorname{e}_0(\mathcal{C})+\lambda({\mathbf R}/\mathcal{C}^{**})\geq 2\lambda({\mathbf R}/\mathcal{C}).$$

Here $\operatorname{cdeg}({\mathbf R})$ and $\operatorname{bideg}({\mathbf R})$ denote the canonical degree and bidual canonical degree, respectively. The conjecture asks whether the canonical degree always dominates the bidual canonical degree in dimension one; the supplied text gives no resolution.