Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number theory) numbers as even integers $ n $ such that $ n $ can be written in at least $ n $ ways as the difference between consecutive odd primes. For example $ 2=13-11=7-5$ is a Sergei number.

Are there provably infinitely many Sergei numbers ? What about the best reachable lower bound of their density among even numbers ?