I managed to track down the article that Reich mentions (S. Rasmussen et al., "Early Divergent Strains of Yersinia pestis in Eurasia 5,000 Years Ago", Cell 163 (2015): 571-82. At the end of the article, the authors write "These early plagues may have been responsible for the suggested population declines in the late 4th millennium BC and the early 3rd millennium BC" and "In light of our findings, it is plausible that plague outbreaks could have facilitated—or have been facilitated by—these highly dynamic demographic events" (meaning the large-scale population movements of the Bronze Age).

The underlying problem is the fact that the lower-density Yamnaya population was able to largely replace - not just conquer - the higher-density farming population. The hypothesis concerning immunity offers an explanation; if the die-off rates had been similar, the explanation fails.