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

A topological group $(G,*)$ is Hurewicz bounded when it satisfies ${\rm S}_1(\Omega_{\rm nbd},\mathcal O^{\rm gp})$. The principle ${\rm S}_c(\mathcal O_{\rm nbd},\mathcal O)$ concerns neighborhood covers, while ${\rm S}_c(\mathcal O,\mathcal O)$ concerns open covers. The conjecture. There is a metrizable Hurewicz 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 is the Hurewicz-bounded weakening proposed in the source; its status is unresolved there.