What's the status of Babai's Graph isomorphism result?
Aggregating comments by  Thomas Klimpel, Sasho Nikolov and  Mohammad Al-Turkistany into a community answer:

The correction (and hence the quasi-polynomial result) was immediately supported by Harald Andrés Helfgott. His expository paper (https://arxiv.org/abs/1701.04372) and its translation (https://arxiv.org/abs/1710.04574) are all the support that is needed for the quasi-polynomial results. Helfgott’s exposition appeared as Exposé 1125 in Astérisque 407 (2019), Séminaire Bourbaki 2016/2017, pp. 135–182.

There are no current known issues with Babai’s proof, and it has gone through extensive peer review. That said, even published peer-reviewed papers have later been withdrawn as incorrect. 

The support by Harald Andrés Helfgott along with the lack of other issues should be sufficient for us to accept that graph isomorphism is indeed solvable in quasi-polynomial time.