The dense subgroup density conjecture for compact groups

Let $G$ be a compact topological group. A subgroup $H\leq G$ is dense if its closure in $G$ is $G$, and $d(H)$ denotes the density character of $H$, while $w(G)$ denotes the weight of $G$. Dense subgroup density conjecture. Every compact group $G$ has a dense subgroup $H$ such that

$$d(H)=w(G).$$

This would strengthen the preceding results for compact abelian and connected compact groups. The statement is presented as a stronger statement that the authors suspect to be true; no resolution is given in the source.