Greenberg's class number conjecture for p-adic towers

Let $p$) be a fixed prime, let $d$ be a positive integer, and let

$$K=K_0\subset K_1\subset K_2\subset\cdots\subset K_n\subset\cdots\subset K_{\infty}$$

be a $\mathbb{Z}_p^d$-extension tower in which $K_n/K$ corresponds to the subgroup $(p^n\mathbb{Z})^d$. Write $\operatorname{Cl}(K_n)$ for the ideal class group of $K_n$. Greenberg's class number conjecture. There exists a polynomial $f(U,V)\in\mathbb{Q}[U,V]$ with total degree $\deg f\leq d$ and degree in $V$ $\deg_V f\leq 1$ such that, for all sufficiently large $n$,

$$v_p(\#\operatorname{Cl}(K_n))=f(p^n,n).$$

Equivalently, there exist non-negative integers $\mu,\lambda,\mu_1,\ldots,\mu_{d-1},\lambda_1,\ldots,\lambda_{d-1}$ and $\nu\in\mathbb{Q}$ such that, for all sufficiently large $n$,

$$v_p(\#\operatorname{Cl}(K_n))=(\mu p^n+\lambda n)p^{(d-1)n}+\sum_{i=1}^{d-1}(\mu_i p^n+\lambda_i n)p^{(d-i-1)n}+\nu.$$

This is a higher-dimensional analogue of Iwasawa's class number formula for $\mathbb{Z}_p^d$-extensions. The statement is presented as Greenberg's conjecture in the source; its resolution status is not specified here.