Are there infinitely many primes of the form $9m^2+3m+1$ where $m\in \mathbb{Z}$?

Question

This question is related to Theorem $1$ in
my old answer.

All primes $p=9m^2+3m+1$ with $m\in \mathbb{Z}$ up to $1000$ can be found in this paper (see Table $1$ on the page $7$).

Answer 1

The Bunyakovsky conjecture implies that there are infinite many primes of this form, but the only solved case is degree $1$ (Dirichlet's theorem).

For no integer polynomial with degree $d>1$, it is known that it produces infinite many primes.