The hyperplane-section inequality for bidual canonical degree

Let $({\mathbf R},{\mathfrak m})$ and $x$ satisfy the conditions above: $\mathcal{C}$ is a canonical ideal of ${\mathbf R}$, $x$ is regular modulo $\mathcal{C}$, ${\mathbf S}={\mathbf R}/(x)$, and $\mathcal{D}=(\mathcal{C},x)/(x)$ is the induced canonical ideal of ${\mathbf S}$.

Hyperplane-section conjecture. Under these conditions,

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

This is a Lefschetz-type comparison predicting that bidual canonical degree does not increase under the indicated hyperplane section. The text presents it as a model for the desired behavior and gives no resolution.