Primality conjecture for indecomposable rigid dual semicanonical basis elements

Let $C$ be a symmetric generalized Cartan matrix, let $\mathfrak{g}=\mathfrak{n}_-\oplus\mathfrak{h}\oplus\mathfrak{n}$ be its associated Lie algebra, and let ${\mathcal S}^*$ be the dual semicanonical basis of the graded dual $U(\mathfrak{n})^*_{\rm gr}$. For an irreducible component $Z$ of a variety of nilpotent preprojective-algebra modules, call $Z$ indecomposable if it contains a Zariski-dense subset of indecomposable modules and rigid if it contains a module $M$ with $\operatorname{Ext}^1_\Lambda(M,M)=0$; write $\rho_Z\in{\mathcal S}^*$ for the corresponding basis element. Primality conjecture. If $\rho_Z\in{\mathcal S}^*$ is indecomposable and rigid, then $\rho_Z$ is prime in $U(\mathfrak{n})^*_{\rm gr}$. The claim would identify all indecomposable rigid dual semicanonical basis elements as prime; the source gives no evidence resolving it.