1. Simply testing something 1000 times and (presumably) presenting the arithmetic mean is not very informative. Looking at the detailed reported benchmark times (in the output file in tests), it looks like many of the timing outcomes have high variance. Rather than running the tests 1000 times and taking the mean, you might consider running 10 batches of 100 tests (or 1000, if you can) and presenting the mean and variance of the resulting distribution. In general, k sample groups each of size p will provide more reliable information about the underlying distribution than one sample group of size k*p (for reasonable k and p, obviously).
2. Related to that, the results of the "inserting a number of elements" and "deleting a number of elements" tests are significantly worse for the tiered vector vs the std::vector than the "insert/delete a single element" tests. You don't mention this in the readme, but thinking about why it is might be informative. Thrashing seems like a possible explanation, and one you might be able to mitigate.
3. Are you making sure your cache is warm before starting to measure performance? (Pardon, I didn't look through every line of your tests.) Particularly for std::vector, and likely your intermediate deques too, this will have a big effect on timing.
4. Finally, it looks like you're primarily testing using ints (?). It would probably be a good idea to see if your results hold for a different payload size.
I don't know whether these will improve or worsen your comparison against std::vector, but they will make your claims more robust.