Generalized Gross conjecture for cyclotomic coinvariants

Let $k/\mathbb{Q}$ be a finite extension and $p$ a prime number. Let $k^\mathrm{c}_\infty/k$ be the cyclotomic $\mathbb{Z}_p$-extension, and let $X'(k^\mathrm{c}_\infty)$ be the Galois group of its maximal unramified abelian pro-$p$-extension in which every $p$-adic prime splits completely. Let $X'(k^\mathrm{c}_\infty)_{\operatorname{Gal}(k^\mathrm{c}_\infty/k)}$ denote the coinvariants.

Generalized Gross conjecture. The coinvariant group $X'(k^\mathrm{c}_\infty)_{\operatorname{Gal}(k^\mathrm{c}_\infty/k)}$ is finite.

This conjecture concerns the finiteness of cyclotomic coinvariants in the setting of completely split $p$-adic primes and is open in general.