Bounded Gaps Between Primes of a Special Form

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