The primary decomposition conjecture for ideals in acyclic cluster algebras

Let $R$ be the ring under consideration, let $I_1,\ldots,I_n$ be its associated ideals, and let $\mathbf{a}=(a_1,\ldots,a_n)\in\mathbb{N}^n$. Write

$$I^{\mathbf{a}}=I_1^{a_1}I_2^{a_2}\cdots I_n^{a_n}.$$

Here $I_1^{a_1}\cap\cdots\cap I_n^{a_n}$ denotes the intersection of the corresponding ideal powers.

Primary decomposition conjecture. For all $\mathbf{a}\in\mathbb{N}^n$,

$$I^{\mathbf{a}}=I_1^{a_1}\cap I_2^{a_2}\cap\ldots\cap I_n^{a_n}.$$

This conjecture asserts that the product of these ideal powers equals their intersection, extending the familiar equality for coprime ideals to the ideals arising in the cluster-algebra description. Its status is not resolved in the supplied source context.