Idempotent conjecture for quandle rings of free products

Let ${X_i}_{i i I}$ be a family of quandles, and suppose that each integral quandle ring $\mathbb{Z}[X_i]$ has only trivial idempotents. Let $\star_{i i I}X_i$ denote their free product quandle. Free-product idempotent conjecture. The quandle ring $\mathbb{Z}[\star_{i i I}X_i]$ has only trivial idempotents. The claim is motivated by results for free quandles and certain free-product constructions, but remains conjectural in the stated generality.