Jorgensen–Leuschke's canonical-module Betti-number conjecture

Let $R$ be a Cohen–Macaulay local ring with canonical module $\omega$. Jorgensen–Leuschke conjecture. The inequality

$$\beta_{1}^{R}(\omega)\leq\beta_{0}^{R}(\omega)$$

implies that $R$ is Gorenstein. The paper states this as an open problem posed by Jorgensen and Leuschke and proves it for idealization rings.