Embedding invariance of the Dyck tilings tiles statistic

Let $\lambda_1$ and $\lambda_2$ be Dyck paths whose corresponding trees are isomorphic when their embeddings in the plane are ignored. For a Dyck tiling $T\in\mathcal{D}(\lambda,*)$, let $\operatorname{tiles}(T)$ denote its number of tiles. Embedding-invariance conjecture. The tiles generating function is invariant under changing the planar embedding of the corresponding tree:

$$\sum_{T\in\mathcal{D}(\lambda_1,*)} t^{\operatorname{tiles}(T)} = \sum_{T\in\mathcal{D}(\lambda_2,*)} t^{\operatorname{tiles}(T)}.$$

The analogous invariance is already established in the paper for the area and disorder statistics, while invariance of the tiles statistic is proposed as a conjecture.