The hyperplane-section conjecture for the bi-canonical degree

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring with a canonical ideal $\mathcal{C}$. Let $x$ be an element of $R$ that is regular modulo $\mathcal{C}$, and suppose that $\mathcal{C}$ is equimultiple. Bi-canonical degree hyperplane-section conjecture. One should have

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

This is a proposed change-of-rings statement for the bi-canonical degree, paralleling the proved inequality for the canonical degree under the same equimultiplicity and regularity hypotheses. Its general validity remains open in the supplied text.