Dimension conjecture for the deformed ring of a type A cluster quiver

Let $n$ be a positive integer. Define the commutative ring $\mathbf{M}^{\text{def}}(n)$ generated by variables $\mathsf{S}_i$ for $i\in[n]$ and $\mathsf{S}_\alpha$ for $\alpha\in\Phi_{>1}$, modulo the right half of the relations in, all relations in, and the relations $\mathsf{S}_i^2=\mathsf{S}_i$. Let $c_{n+1}$ denote the Catalan number indexed by $n+1$. Dimension conjecture. The ring $\mathbf{M}^{\text{def}}(n)$ has dimension $c_{n+1}$. The deformation replaces the square-zero relations of the original presentation by idempotent relations, so the resulting ring is no longer graded; the conjectured Catalan-number dimension describes the expected size of this deformed algebra.