Greenberg's conjecture for totally real fields

Let $k$ be a totally real number field and $p$ a prime number. For the cyclotomic $bZ_p$-extension $k^\mathrm{c}_\infty/k$, let $X(k^\mathrm{c}_\infty)$ denote its unramified Iwasawa module.

Greenberg's conjecture. The module $X(k^\mathrm{c}_\infty)$ is finite.

This is the classical finiteness conjecture in Iwasawa theory for totally real number fields; its status is open in general.