Cyclic decomposition conjecture for Whittaker-induced modules

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $Z(\mathfrak{g})$ be its center, and let $U$ be a finite-dimensional $Z(\mathfrak{g})$-module. Let $I_\eta(U)$ denote the corresponding Whittaker-induced module. Cyclic decomposition conjecture. Every finite-dimensional $Z(\mathfrak{g})$-module is isomorphic to a direct \sum of cyclic $Z(\mathfrak{g})$-modules. Moreover, every module of the form $I_\eta(U)$ is isomorphic to a direct \sum of cyclic $U(\mathfrak{g})$-modules. The source itself flags the assertion as probably false because $Z(\mathfrak{g})$ is not a principal ideal domain when the rank of $\mathfrak{g}$ exceeds one, so its status should be checked as a likely refutation rather than treated as established.