When is $p^2+1$ twice of a prime?

We have infinitely many primes of the form $$\frac{p^2+1}{2}$$ where $p$ is itself prime, if we have infinite many positive integers $k$ such that $$2k+1$$ and $$2k^2+2k+1$$ are simultaneously prime. (in this case, just set $p=2k+1$). The Bunyakovsky conjecture implies that this is the case, so very likely infinitely many examples exist. But I am convinced that the problem is open.