Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms $n\cdot2^{n+a}+b$.

It seems evident that the same result would hold without the $n$ out front, that is, for $a,b,c$ with $a>0$ and $b>1$ there are only $o(x)$ primes of the form $a\cdot b^n+c$ with $n\le x.$ Has this been proved?

This is essentially the same as a question asked on math.se which was never answered.