Upper-bound intersection conjecture for noncommutative polygons

For each $n\ge 2$, let $\mathcal{U}_n$ be the subalgebra generated by all $x_{ij}$, $i\ne j$, and $x_{i,i^\pm}^{-1}$, and let $\mathcal{A}_\Delta$ be the algebra associated with a triangulation $\Delta$ of $[n]$. Upper-bound intersection conjecture.

$$\mathcal{U}_n=\bigcap_{\Delta}\mathcal{A}_\Delta,$$

where the intersection is over all triangulations $\Delta$ of $[n]$. This is proposed as a totally noncommutative analogue of the upper cluster algebra of type $A_{n-3}$; the source gives no resolution.