I have heard that rumor before but don't believe it to be true. For HBS, I don't think there much of an advantage between R1 and R2 (unlike some other schools like Stanford, where it greatly increases your odds to apply R1) -- I think the difference in acceptance rate by round is more a product of self-selection, with more competitive applicants from traditional backgrounds ready to submit in R1 and more applicants throwing in an application to HBS as a reach school in R2. I suppose if you have lower stats and are being compared in R1 to bankers with 760's that might theoretically put you at a disadvantage, but HBS looks at the application holistically and compares you to others in your bucket (e.g. other entrepreneurs from the U.S.), so it's more important how you stack up against others who applied that year.
MBA Data Guru wrote a good post that addresses the percentage of students admitted in R1 (nearly 12%) vs. R2 (only 6%):
https://www.mbadataguru.com/blog/admissi ... -analysis/Bottom line -- apply when you've put the best application as possible together (but do not apply R3). There may be a *slight* advantage to R2 for certain types of applicants, but in general, you have a higher chance of getting admitted R1.