A metrizable Menger bounded group with a Menger bounded square lacking neighborhood selection

Let $(G,*)$ be a metrizable topological group, and let $G^2=G\times G$ denote its square. The conjecture. There is a metrizable Menger bounded group $(G,*)$ with property

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

such that $G^2$ is Menger bounded but does not have

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

This conjecture asserts that Menger boundedness does not suffice for the product conclusion of the preceding theorem; the source gives no resolution.