Let $\mathcal O_{\mathbb K}$ be the ring of integers in a number field $\mathbb K$. It is easy to prove that the number of (non-associated) primes in $\mathcal O_{\mathbb K}$ is infinite. On the other hand, for any field $\mathbb F$ there is only one (non associated) prime in the power series ring $\mathbb K[[x]] $, namely the element x. Any integral domain that I imagine the question of finitude or infinity of primes is easy.

My question is: is there any integral domain in which the question of finitude or infinitude of prime elements is open? or else very difficult to prove?

Let $\mathcal O_{\mathbb K}$ be the ring of integers in a number field $\mathbb K$. It is easy to prove that the number of (non-associated) primes in $\mathcal O_{\mathbb K}$ is infinite. On the other hand, for any field $\mathbb F$ there is only one (non-associated) prime in the power series ring $\mathbb F[[x]] $, namely the element x. Any integral domain that I imagine the question of finitude or infinity of primes is easy.

My question is: is there any integral domain in which the question of finitude or infinitude of prime elements is open? Or, at least, very difficult to prove?