Primes dividing $x^2+xy+y^2+1$ 

Is there infinitely prime numbers $p$ such that $p$ divides  $x^2+xy+y^2+1$ for some integers $x,y \in \Bbb Z$ ?


I can show that every prime $p$ divides $x^2+y^2+1$ for some integers $x,y \in \Bbb Z$, see Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $. :
the sets $\{-1-x^2 \mid x \in \Bbb F_p\}$ and $\{y^2 \mid y \in \Bbb F_p\}$ have both cardinality $(p+1)/2$, so they must have a non-empty intersection, i.e. an element of the form $z=-1-x^2=y^2$.
But the proof does not work to show that all but finitely many (or at least infinitely many) primes divide $x^2+xy+y^2+1$ for some integers $x,y$. So what can I do?

If $p\ne2$ then $p\mid(x^2+xy+y^2+1)$ iff $p\mid(4x^2+4xy+4y^2+4)$
that is iff $p\mid((2x+y)^2+3y^2+4)$. The same trick as before proves that there
are $y$ and $z$ with $p\mid(z^2+3y^2+4)$.
