A metrizable Menger bounded group separating neighborhood and open-cover selection

A topological group $(G,*)$ is Menger bounded when it satisfies the relevant neighborhood selection property, and {\rm S}_c( obreak\text{\mathcal O}_{\rm nbd},\mathcal O) and ${\rm S}_c(\mathcal O,\mathcal O)$ are the corresponding selection principles for neighborhood and open covers. The conjecture. There is a metrizable Menger bounded topological group which has the property

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

but not the property

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

This asks whether Menger boundedness can replace the Hurewicz property in the preceding separation result; the source gives no resolution.