Greenberg's analogue for S-ramified extensions of CM fields

Let $p$ be an odd prime, let $K$ be a CM field, and let $K^+$ be its maximal totally real subfield. Let $S^+$ be a set of primes of $K^+$ above $p$, each of which splits in $K$ as $\mathcal{P}_i\mathcal{O}_K=\mathfrak{P}_i\widetilde{\mathfrak{P}}_i$, and set

$$S=\\{\mathfrak{P}_1,\mathfrak{P}_2,\dots,\mathfrak{P}_s\\}.$$

Assume Leopoldt's conjecture for $K$, so that the $S$-ramified $\mathbb{Z}_p$-extension $K_\infty/K$ unramified outside $S$ is unique. If $A_n$ denotes the $p$-primary part of the class group of the $n$-th layer of $K_\infty/K$, then the analogue of Greenberg's conjecture. The groups $A_n$ remain bounded as $n\to\infty$; equivalently, the Iwasawa invariants satisfy $\mu=\lambda=0$. The conjecture is proposed as an analogue of Greenberg's conjecture for cyclotomic extensions of totally real fields, motivated by numerical evidence for imaginary quadratic fields; no resolution is given in the source.