Finiteness conjecture for logarithmic class groups of real quadratic fields

Let $K$ be a given real quadratic field. For a prime $\ell$, let the $\ell$-group of logarithmic classes of $K$ be the $\ell$-primary subgroup of its logarithmic class group.

Finiteness conjecture. There are only finitely many primes $\ell$ for which the $\ell$-group of logarithmic classes of $K$ is nontrivial.

This conjecture contrasts with the heuristic expectation, discussed for imaginary quadratic fields, that infinitely many such primes may occur. The supplied text gives no resolution, so the conjecture remains open.