Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$? I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^2-a}{2}+b^4\tag{1}$$
where $a$ and $b$ run over positive integers.

Question. Is there any reasoning or heuristic to get an idea about the veracity of previous conjecture? Many thanks.

Remarks and motivation. The first few terms of previous expression $(1)$ are  $$2, 13, 17, 23, 67, 71,\ldots$$
I believe that this sequence isn't in the OEIS. My motivation to ask about previous conjecture was to propose a variation of the work by Friedlander and Iwaniec, involving the pentagonal numbers (a difererent set of polygonal numbers) instead of perfect squares $a^2$.
References:
[1] The article Friedlander–Iwaniec theorem from the encyclopedia Wikipedia.
 A: What can be done, which is already remarked in the cited paper by Friedlander and Iwaniec, is that for any positive definite binary quadratic form $f$ which has no local obstructions, there ought to exist infinitely many pairs $(a,b) \in \mathbb{Z}^2$ such that $f(a,b^2)$ is prime. Even though this is widely believed, the technical weight of the F-I paper has thus far prevented anyone from writing this up formally and publishing it. 
As remarked by Wojowu, the crucial property of the polynomial $x^2 + y^4$ is that it factors over a number field. This additional structure enables one to obtain satisfactory bilinear sum estimates, which Friedlander and Iwaniec shows in a companion paper to the above cited paper is all that's needed for Bombieri's sieve to produce primes. So far, the existence of a norm form structure (the only polynomials we know that 'obviously' produce infinitely many primes) is necessary to obtain such bilinear sum estimates.
Aside from the Friedlander-Iwaniec paper mentioned, there are three seminal papers that run along these lines:
1) D.R. Heath-Brown, Primes represented by $x^3 + 2y^3$, Acta Math., (1) 186 (2001), 1-84.
2) J. Maynard, Primes represented by incomplete norm forms, https://arxiv.org/abs/1507.05080.
3) D.R. Heath-Brown, X. Li, Prime values of $a^2 + p^4$, Invent. Math (2) 208, 441-499.
In general, one would expect that any polynomial that is not reducible over $\mathbb{Q}$ and for which there are no obvious obstructions would represent infinitely many primes. This is considered well beyond reach of current methods. 
