The authors sequenced the genome of 120 individuals representing all of Darwin’s finches. They found that a 240 kilobase haplotype encompassing the ALX1 gene, which encodes a transcription factor affecting craniofacial development, is strongly associated with beak shape diversity across Darwin’s finch species, as well as the medium ground finch (Geospiza fortis), a species that the Grants observed undergoing rapid evolution of beak morphology in response to the environmental changes described in their 2006 paper.
The authors saw variants of this gene in the finches, each of which encoded for a different type of beak morphology. These variants had arisen during natural selection processes.

Order of authorship in the original publication: Lamichhaney, Berglund, .... Grant, Grant.

(Thanks to Juan Weiss for this reference and its interpretation.)