A note on large clusters of $E_2$-numbers

Let $q_n$ denote the $n$th product of two distinct primes. By completing a square in Sono’s sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every $\epsilon >0$ and all sufficiently large integers $\rho$, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}(q_{n+\rho }-q_n)\le \exp ((\pi +\epsilon )\sqrt{\rho })\mathrm{.}$$