Polignac's conjecture (PC) is that there exists infinitely many pairs of consecutive prime numbers that are a distance $d$ apart for some natural number $d$. The twin prime conjecture is the particular instance of this conjecture for $d = 2$. The fact that this conjecture remains open has some interesting implications on nonstandard models of Peano Arithmetic (PA). Specifically, it is a standard exercise to show that every model of PA has order type $\mathbb{N} + \mathbb{Z} \cdot A$ for some dense linear order without endpoints. Thus every nonstandard model has an initial segment of Natural numbers followed by nonstandard numbers all appearing in an unbounded dense linearly ordered collection of what are called integer blocks or $\mathbb{Z}$-blocks. What I realized (someone else must've realized this too so please mention references if you know of any) is that if Polignac's conjecture turned out to be false, then we'd have the following strong limitation on the number of primes appearing in $\mathbb{Z}$-blocks.

($\mathbb{N} \vDash \lnot PC$) If $M$ is a model of PA and is $\Sigma^0_1$-equivalent to the theory of $\mathbb{N}$, then $M$ can have at most one prime appearing in any $\mathbb{Z}$-block: If not, then for some $d \in \mathbb{N}$, there would be a pair of nonstandard numbers that $M$ would view as two consecutive primes a distance $d$ apart. Since this occurs in a $\mathbb{Z}$-block, for any true Natural number $n$, the model $M$ thinks that there is a pair of consecutive primes greater than $n$ a distance $d$ apart. Then by $\Sigma^0_1$-elementarity, $\mathbb{N}$ would think the same thing so $\mathbb{N}$ would have unboundedly many pairs of consecutive primes a distance $d$ apart, making Polignac's conjecture true (in the standard model).

My question concerns the other models of PA:

Can we prove that there is a nonstandard model of PA having a $\mathbb{Z}$-block with at least two primes? Even better, can we prove that there is a model of PA with unboundedly many $\mathbb{Z}$-blocks having at least two primes?