Is it true that for all large enough integer $n$, there is a integer $m$ such that $n+im$ is a Gaussian prime?

Question

Let $n$ denote a positive integer greater than $6$. Is it known whether there is always an integer $m$ such that $n+im$ is a Gaussian prime?

Answer 1

This statement is equivalent to :

For every large enough positive integer $n$, there is a positive integer $m$ such that $m^2+n^2$ is prime.

This follows from the Bunyakovsky conjecture , but as far as I know it is unknown whether this is the case.