The Mastodon paleogenomics paper is out on PLOS:
We obtained the sequence from a tooth dated to 50,000-130,000 years ago, increasing the specimen age for which such palaeogenomic analyses have been done by almost a complete glacial cycle. Using this sequence, together with mitochondrial genome sequences from two African elephants, two Asian elephants, and two woolly mammoths (all of which have been previously sequenced), we show that mammoths are more closely related to Asian than to African elephants. Moreover, we used a calibration point lying outside the Elephantidae radiation (elephants and mammoths), which enabled us to estimate accurately the time of divergence of African elephants from Asian elephants and mammoths (about 7.6 million years ago) and the time of divergence between mammoths and Asian elephants (about 6.7 million years ago). These dates are strikingly similar to the divergence time for humans, chimpanzees, and gorillas, and raise the possibility that the speciation of mammoth and elephants and of humans and African great apes had a common cause. Despite the similarity in divergence times, the substitution rate within primates is more than twice as high as in proboscideans.
I don't know much about paleobiology, but the idea that there might be correlated speciation events across multiple taxa sounds like the Turnover Pulse Hypothesis.