TITLE: Is every prime the largest prime factor in some prime gap?
QUESTION [19 upvotes]: 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, 21 Dec 2019: Conjecture verified for $p \le 10^{10}.$
Note: This question was posted in MSE and got many upvotes but no answer hence posting in MO.

REPLY [3 votes]: Extra long comment sharing data as requested by @GerhardPaseman
Here is the data for all primes $\le 16290041$. The way to read this data is as follows: Taking the first row as an example, of the $1048575$ primes $\le 16290041$, there are exactly $774792$ primes $p$ such that the integer $2p$ causes the largest prime factor in the prime gap containing it to be $p$, i.e. nearly $74\%$ of the times. And so on for the rest of the rows.
This nearly $26\%$ of the times we need a multiple $k > 2$.