Completely split primes in non-anticyclotomic $\mathbb{Z}_p$-extensions

Question

In his colloquium paper "The Structure of Selmer Groups" Greenberg writes the following:

If $K$ is an imaginary quadratic field ... it is conjectured that for any [non-anticyclotomic] $\mathbb{Z}_p$-extension of K at most one prime of $K$ can split completely. (One can prove that at most two can.)

(end of the first paragraph on page 3)

I am looking for a reference in which this conjecture is formulated and also where the claim in parentheses is proven. I have searched through most, if not all, of the author's articles and much of the existing literature but have not been able to find anything regarding this question. Perhaps the conjecture is folklore or it follows easily from some other more well-known conjecture that I have not considered.

A further question after this would be if there is anything one conjectures (or can even prove) in the general CM case?

Thanks in advance!

Answer 1

The article Sur les ideaux dont l'image par l'application d'Artin dans une $\mathbb{Z}_p$-extension est triviale of Michel Emsalem provides a satisfactory answer to the general question of how many places can split in a $\mathbb{Z}_p$-extension.

As such, I will mark this question as answered.