The infinitude conjecture for imaginary quadratic fields with minimal absolute abelian Galois group

Let $K$ be an imaginary quadratic field, and let $G$ denote the profinite group

$$G={\widehat{{\mathbf{Z}}}}^2 \times \prod_{n\ge1} {\mathbf{Z}}/n{\mathbf{Z}}.$$

Infinitude conjecture. There are infinitely many imaginary quadratic fields $K$ for which the absolute abelian Galois group is isomorphic to $G$.

The group $G$ is the minimal Galois group identified in the surrounding discussion, and the conjecture is motivated by computational evidence from imaginary quadratic fields of small odd prime class number. The source gives no resolution, so the conjecture remains open.