Is the question of existance of infinitely many primes of the form $n^2+1$ equivalent to existance of infinetely many twin primes in ring of Gaussian integers $\mathbb{Z}[i]$?
By twin prime Gaussian integers I mean prime ones with smallest possible difference (i.e. $2i$ or $2$).
Note that if $n^2+1$ is prime then $(n+i)(n-i)$ is prime in $\mathbb{Z}$, so from the fact that $\overline{n+i} = n - i$ we can get that both $n+i$ and $n-i$ are prime in $\mathbb{Z}[i]$.
EDIT: From the comments below I think it would be better to reformulate my question to
Which natural conjecture about Gaussian primes imply "$n^2+1$" conjecture?