Weak form of Greenberg's conjecture

Let $k$ be a totally real number field and $p$ a prime number. Let $k^\mathrm{c}_\infty/k$ be its cyclotomic $bZ_p$-extension, and let $X(k^\mathrm{c}_\infty)$ be the associated unramified Iwasawa module. A finite $bZ_p\llbracket\operatorname{Gal}(k^\mathrm{c}_\infty/k)\rrbracket$-submodule means a submodule that is finite as an abelian group.

Weak form of Greenberg's conjecture. If $X(k^\mathrm{c}_\infty)$ is nontrivial, then it has a nontrivial finite $bZ_p\llbracket\operatorname{Gal}(k^\mathrm{c}_\infty/k)\rrbracket$-submodule.

This is presented as a weaker variant of Greenberg's conjecture and is open in the generality stated.