A conjecture on consecutive odd composite numbers

Question

Can you provide a proof or a counterexample for the claim given below?

Inspired by Grimm's conjecture I have formulated the following claim:

Let $n_1,n_2,\dots,n_k$ be a sequence of $k$ consecutive odd numbers which are all composite. Let $\operatorname{gpf}(n_i)$ be the greatest prime factor of $n_i$. Then, all $\operatorname{gpf}(n_i)$, $1 \le i \le k$ are mutually different.

Try it yourself.

Answer 1

A conterexample: gpf(20449) = gpf(20475) = 13, and there are no prime numbers between them. According to my program's results, it is the only counterexample up to 1000000.