definition: In the gap between two consecutive odd prime numbers, we have one or more compound numbers. We define the largest of all prime factors of this composition as the maximum prime factor of the gap.
claimEvery prime number is a maximum prime factor for a prime number gap.
I am looking for a proof or a proof.
Update, December 7, 2019: Guess confirmed for $ p le 2 times 10 ^ 9. $
Note: This question was posted in MSE and received many positive votes, but no answer, so it was posted in MO.