Walk enumeration conjecture for square non-zero recurrent configurations

Let $D_{n,n}$ be the relevant complete bipartite graph with one sink, let $\mathsf{Rec}^{\star}(D_{n,n})$ denote its recurrent configurations, and write $u_1,\ldots,u_{m+n-1}$ for the heights at the non-sink vertices. Define

$$a_n=\left\{u\in\mathsf{Rec}^{\star}(D_{n,n}):u_1,\ldots,u_{m+n-1}>0\right\}.$$

Square non-zero configuration conjecture. The cardinality satisfies

$$a_n=\frac{1}{n-1}\binom{2n-2}{n}\binom{2n}{n-2},$$

and this is also the number of walks from $(0,0)$ to $(0,1)$ that remain in the upper half-plane $y\geq0$ and use $2n-3$ unit steps from $\{\mathsf{n},\mathsf{s},\mathsf{e},\mathsf{w}\}$.

The conjecture proposes a closed enumeration of square non-zero recurrent configurations and identifies the same numbers with constrained planar walks. The paper gives no proof or resolution beyond stating the conjecture.