Schwarz (1999) reviews a variety of research on frequency estimation scales. That's not exactly the same kind of scale, but both request estimation of a count variable by choice among ordered, polytomized response options. Spector (1976, 1980, 1992) may also have something useful to say on this topic: he researched the effect of providing even vs. uneven interval categories for rating scales of frequency. Meanwhile, Moskowitz and a different Schwarz (1982) have compared the reliability and validity of behavior counts and ratings in personality research.
Getting somewhat off-topic, culture may be important to consider for Likert scale response interpretation in general (see for example Lee, Jones, Mineyama, & Zhang, 2002).
Lee, J. W., Jones, P. S., Mineyama, Y., & Zhang, X. E. (2002). Cultural differences in responses to a Likert scale. Research in Nursing & Health, 25(4), 295–306.
Moskowitz, D. S., & Schwarz, J. C. (1982). Validity comparison of behavior counts and ratings by knowledgeable informants. Journal of Personality and Social Psychology, 42(3), 518–528.
Schwarz, N. (1999). Self-reports: How the questions shape the answers. American Psychologist, 54(2), 93–105. Available online, URL: https://edit.ethz.ch/oat/education/material/material_06_07/material_empirical_methods_06_07/American_Psychologist_Self-Reports.pdf.
Spector, P. E. (1976). Choosing response categories for summated rating scales. Journal of Applied Psychology, 61(3), 374–375.
Spector, P. E. (1980). Ratings of equal and unequal response choice intervals. The Journal of Social Psychology, 112(1), 115–119.
Spector, P. E. (Ed.). (1992). Summated rating scale construction: An introduction (No. 82). Sage.