Small gaps between almost-twin primes

Let $m\in \mathbb{N}$ be large. We show that there exist infinitely many primes $q_1<\cdot \cdot \cdot <q_{m+1}$ such that $$q_{m+1}-q_1=O(e^{7.63m})$$ and $q_j+2$ has at most $$\frac{7.36m}{\log 2}+\frac{4\log m}{\log 2}+21$$ prime factors for each $1\le j\le m+1$. This improves the previous result of Li and Pan, replacing $m^4{e}^{8m}$ by $e^{7.63m}$ and $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ by $\frac{7.36m}{\log 2}+\frac{4\log m}{\log 2}+21$. The main inputs are the Maynard-Tao sieve, a minorant for the indicator function of the primes constructed by Baker and Irving, for which a stronger equidistribution theorem in arithmetic progressions to smooth moduli is applicable, and Tao’s approach previously used to estimate ${\sum }_{x\le n<2x}{\mathbf{1}}_{\mathbb{P}}(n){\mathbf{1}}_{\mathbb{P}}(n+12){\omega }_n$, where ${\mathbf{1}}_{\mathbb{P}}$ stands for the characteristic function of the primes and ${\omega }_n$ are multidimensional sieve weights.