Bounded gaps between primes of the special form

For each $m\ge 1$, there exist infinitely many primes $p_1<p_2<\dots <p_{m+1}$ such that $p_{m+1}-p_1=O(m^4{e}^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.