A metrizable group with a Menger square lacking neighborhood selection

Let $(G,*)$ be a metrizable topological group, and write $G^2=G\times G$ for its square. The conjecture. There is a metrizable group $(G,*)$ which has the property

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

and $G^2$ has the Menger property but does not have

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

This is the proposed weakening from Menger boundedness to the Menger property in the product theorem, and it remains unresolved in the source.