Given integers $a_1,\ldots,a_k>0$ and $b_1,\ldots,b_k$, consider the polynomial $f(x) = \prod_{i=1}^{k} (a_i x +b_i) \in \mathbb{Z}[x]$. Suppose that $\{ a_i x+b_i\}_{i=1}^{k}$ are pairwise coprime polynomials of degree 1.
It is known (see e.g. Chapter 10 of `Sieve Methods' by Halberstam and Richert) that if $\gcd(f(1),f(2),f(3),\ldots)=1$, then for any sufficiently large $r$ (in terms of $k$), there are infinitely many $n$-s such that $f(n)$ is a product of at most $r$ primes. In fact, there are at least $\Omega_{a_i,b_i,k,r}(\frac{x}{\log^k x})$ such $n$-s up to $x$, and one may take $r > C k \log (k+1)$ for some absolute constant $C$.
Is it possible to prove a similar result with modular constraints on the prime factors of $F(n)$? E.g., is it known that given $k$, there is an integer $r$ with the following property: $f(x)=\prod_{i=1}^{k} (4 a_i x + 1)$ is a product of at most $r$ primes of the form $1 \bmod 4$ for infinitely many $n$?
Note that the condition that the prime factors of $f(n)$ are $1 \bmod 4$ is consistent with the congruence $f(n) \equiv 1 \bmod 4$.
