Are there any pairs of integers that are divisible by the same primes such that adding $1$ or $2$ also keeps them divisible by the same primes?

Question

The answer to this question shows that there are infinitely many pairs of integers $(m, n)$ such that $m$ and $n$ have the same prime factors, and $m+1$ and $n+1$ also have the same prime factors. Are there any such pairs of integers with that have the same prime factors such that $m+1$ and $n+1$ have the same prime factors, and $m+2$ and $n+2$ also have the same prime factors? If so, are there infinitely many such pairs?