The hyperplane-section inequality for bi-canonical degree

Let $({\mathbf R},{\mathfrak m})$ be a Cohen–Macaulay local ring with canonical ideal $\mathcal{C}$. Let $x$ be regular modulo $\mathcal{C}$, and suppose that $\mathcal{C}$ is equimultiple. Set ${\mathbf S}={\mathbf R}/(x)$. The image of $\mathcal{C}$ in ${\mathbf S}$ is a canonical ideal, denoted by $\mathcal{D}=(\mathcal{C},x)/(x)$.

Hyperplane-section conjecture.

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

Equivalently, with $\mathcal{D}$ as above, the proposed inequality compares $\operatorname{bideg}({\mathbf R})=\deg({\mathbf R}/\mathcal{C})-\deg({\mathbf R}/\mathcal{C}^{**})$ with $\operatorname{bideg}({\mathbf S})=\deg({\mathbf S}/\mathcal{D})-\deg({\mathbf S}/\mathcal{D}^{**})$. The source presents this as a desirable comparison modeled on the corresponding canonical-degree inequality; no resolution is given.