Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other number in the gap. E.g. $43$ is the largest prime in the gap between the consecutive primes $83$ and $89$. I am interested in the largest prime factor in the gap between two consecutive primes.

Claim: Every prime is the largest prime factor in some prime gap.

I am looking for a proof or disproof.

Update, 7 Dec 2019: Conjecture verified for $p \le 6 \times 10^9.$

Note: This question was posted in MSE and got many upvotes but no answer hence posting in MO.