Are they twin primes?

Question

Let $n$ be a positive integer and let $p_1, p_2,...,p_n$ be first n primes. And let $m$ be the smallest integer $m\ge\,p_n$ such that $m$ and $m+2$ are coprime to $p_1,p_2,..., p_n$.

Are $m$ and $m+2$ always prime?