Indecomposability conjecture for cyclotomic quiver Hecke algebras

Let $A=(a_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix, let $\Lambda\in P^+$ be dominant, and let $\beta\in Q_n^+$. Let $\mathscr{R}_{\beta}^{\Lambda}$ be the associated cyclotomic quiver Hecke algebra. Indecomposability conjecture. The algebra $\mathscr{R}_{\beta}^{\Lambda}$ is indecomposable. Equivalently, the degree-zero component of its center is one-dimensional, namely $Z(\mathscr{R}_{\beta}^{\Lambda})_0=K e(\beta)$. The conjecture is stated as a major unsolved problem and is related to the degree-zero center and cocenter of the algebra.