The generalized Landau conjecture for quadratic polynomials

For a positive integer $m$, let $f(x)=mx^2+1$.

Generalized Landau conjecture. The polynomial $f$ represents infinitely many primes.

This is a folklore conjecture and a special case of the Bunyakovsky conjecture. The case $m=1$ is one of Landau's problems and remains open; Iwaniec proved that infinitely many positive integers $n$ make $f(n)$ either prime or a product of two primes.