The most important information in deciding if the birds belong to a single species or to two separate species is:
When individuals from different groups mate, the offspring are sterile.
This information is crucial because it indicates that the two groups are reproductively isolated from each other, which is a key criterion used to define separate species according to the biological species concept. If two groups can mate but produce sterile offspring, it suggests they are distinct species, as they cannot successfully reproduce and produce viable, fertile offspring.