There could be other hypotheses. One would be genetic drift in the species which has an altered bill in the sympatric case. The test there would be to look at allelic frequencies for genes governing bill shape in both populations to check for a significant difference.
Work Step by Step
There are other possible hypotheses as well; scan the book for things like genetics which can lead to differences between populations. The nature of the thing--genetics--directs the kind of test to make, which in this case is measurement of allelic frequencies.