Many complex networks exhibit hierarchical, tree-like structures, making hyperbolic space a natural candidate wherein to learn representations of them. Based on this observation, Hyperbolic Graph Neural Networks (HGNNs) have been widely adopted as a principled choice for representation learning on tree-like graphs. In this work, we question this paradigm by proposing the additional condition of geometry--task alignment, i.e., whether the metric structure of the target follows that of the input graph. We start by empirically showing that HGNNs can recover low-distortion representations when supervision explicitly rewards metric preservation and proceed by theoretically analyzing a controlled regression setting in which a linear readout benefits from representations whose geometry is aligned with the target metric. Finally, by relying on both predictive performance and embedding distortion, we further show that HGNNs gain an advantage on link prediction, a naturally geometry-aligned task, whereas this advantage largely disappears on standard node classification benchmarks. Overall, our findings shift the focus from only asking "Is the graph hyperbolic?" to also questioning "Is the task aligned with hyperbolic geometry?", showing that HGNNs consistently outperform Euclidean models under such alignment, while their advantage vanishes otherwise.

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