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. I am interested in this prime factor which is the largest in this gap.

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 $p \le 2 \times 10^9.$

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