I am interested in having an upper bound for the cardinality of

$\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$ for $k,\ell\geq 1$,

where $\omega(n)=\sum_{p\vert n}1$ counts the number of (distinct) prime factors of the integer $n$.

I would actually like a sharp upper bound (up to a constant) for $k=\ell=\log\log x$.
I thought this was known, but I can't find anything in the literature. For $k\ll\sqrt{\log\log x}$ and $\ell\ll\log\log x$ I can have a good upper bound, but I can't reach $k=\ell=\log\log x$.

Does anyone has a reference or an idea please ?

Thank you very much !