Davidski notes other serious problems with the data analysis in the paper (particularly the assertion I have emphasized in the abstract). But, the rise of lactase persistence is well documented from multiple sources.
I personally think it is more likely that an LP variant was present, but at low frequency in Steppe peoples and then expanded in frequency exponentially due to selection on this variant in Northern Europe. But this certainly isn't the first study to show that LP wasn't common on the steppe and arose from near zero during the Bronze Age in Northwest Europe
The question of why LP was so strongly selected for also remains an open question. There is really no indicated of a strongly punctuated selection for it - it continued more or less steadily for centuries. My best guess would be that the impact on infant morality via the nutritional health of nursing mothers is the most important factor. The fact that steppe people hailed from a lower latitude and might have been prone to Vitamin D deficiencies at a greater rate than the locals could also have been a factor.
Lactase persistence (LP), the continued expression of lactase into adulthood, is the most strongly selected single gene trait over the last 10,000 years in multiple human populations. It has been posited that the primary allele causing LP among Eurasians, rs4988235-A, only rose to appreciable frequencies during the Bronze and Iron Ages, long after humans started consuming milk from domesticated animals. This rapid rise has been attributed to an influx of people from the Pontic-Caspian steppe that began around 5,000 years ago.
We investigate the spatiotemporal spread of LP through an analysis of 14 warriors from the Tollense Bronze Age battlefield in northern Germany (∼3,200 before present, BP), the oldest large-scale conflict site north of the Alps. Genetic data indicate that these individuals represent a single unstructured Central/Northern European population.
We complemented these data with genotypes of 18 individuals from the Bronze Age site Mokrin in Serbia (∼4,100 to ∼3,700 BP) and 37 individuals from Eastern Europe and the Pontic-Caspian Steppe region, predating both Bronze Age sites (∼5,980 to ∼3,980 BP).
We infer low LP in all three regions, i.e., in northern Germany and South-eastern and Eastern Europe, suggesting that the surge of rs4988235 in Central and Northern Europe was unlikely caused by Steppe expansions. We estimate a selection coefficient of 0.06 and conclude that the selection was ongoing in various parts of Europe over the last 3,000 years.