Iwaniec proved (and many people extended) that the number of $n \le x$ for which $n^2+1=P_2$ (product of at most two primes) is $\gg x/\log x$. I am wondering what is known/can be proved for the counting function of a localised version of this problem $$\{n\le x \mathrel\vert\exists p,q\geq x^\alpha\text{ primes}:pq \mid n^2+1\}$$ for $\alpha> 1/2$ (or say $\alpha=1-\varepsilon$).

In particular, is it possible to prove a cheap *upper bound* of the (nearly) correct order of magnitude?

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