Finiteness conjecture for nontrivial logarithmic class groups

Let $k$ be a fixed number field. Assume that $k$ satisfies the Leopoldt and Gross–Kuz'min conjectures for every prime. Let $\widetilde{\mathcal H}_k(p)$ denote the logarithmic class group of $k$ at the prime $p$.

Finiteness conjecture for logarithmic class groups. The set

$$\{p: \widetilde{\mathcal H}_k(p)\ne 1\}$$

is finite.

This predicts that only finitely many primes yield a nontrivial logarithmic class group for a fixed field satisfying the stated universal conjectural hypotheses. The source gives no evidence of resolution.