The Boehm paper says "Fig. 3 shows a summary of current experience at IBM[4], GTE[5], and TRW on the relative cost of correcting software errors as a function of the phase in which they are corrected." and indeed figure 3 (page 1228) shows exponential growth. It only shows averages or ranges for each data source, and that's a legitimate critique. That said, it does show them for multiple companies, and then presents a trend line that plausibly follows from the data provided. Boehm has a good reputation, I expect that this really was a reasonable observation from real data.
It's legitimate to question whether or not that is still true. Computer "science" is notorious for having almost no science - experiments are almost non-existent. I would love to see this & many other experiments conducted to see what's true today.