Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. We define the largest of among all the prime factor of these composite as the maximal prime factor of the gap.

Claim: Every prime is a maximal prime factor for 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.