A metrizable Menger group separating neighborhood and open-cover selection

A topological group is Menger when it satisfies the Menger selection property, equivalently ${\rm S}_{\rm fin}(\mathcal O,\mathcal O)$. The principles ${\rm S}_c(\mathcal O_{\rm nbd},\mathcal O)$ and ${\rm S}_c(\mathcal O,\mathcal O)$ concern neighborhood and open covers, respectively. The conjecture. There is a metrizable topological group which has the Menger property and property

$${\rm S}_c(\mathcal O_{\rm nbd},\mathcal O),$$

but not the property

$${\rm S}_c(\mathcal O,\mathcal O).$$

This is the Menger-property weakening discussed by the authors, and the source provides no resolution.