small change in the luminosity required for a discovery by a factor of about $2$. Note that the discovery curves are the expected discovery reach. Since for ${{r_{\beta}}}=10^{-10}$ the sufficient event number for discovery is one or two, the statistical fluctuation is of $\mathcal{O}(1)$, too. So a discovery can easily take place at half or double the integrated luminosity shown in the figures.[^4] Thus, the difference between the ${{r_{\beta}}}=10^{-7}$ and ${{r_{\beta}}}=10^{-10}$ discovery curve is of the same order as the expected statistical fluctuation. For this reason we assess the discovery curves to be relatively insensitive to the change in ${{r_{\beta}}}$ in the considered range. Moreover, the exclusion limits for ${{r_{\beta}}}=10^{-7}$ and $10^{-10}$ become degenerate at the three-event line. Thus, the exclusion potential is not sensitive to ${{r_{\beta}}}$ at all.

This also sheds some light on the impact of the uncertainties in the background cross section. The variation of ${{r_{\beta}}}$ over three orders of magnitude reveals how little impact an uncertainty of tens or even hundreds of percent in the background cross section would have. This justifies the somewhat arbitrarily chosen $K$-factor for the muon background in section \[sec:background\] and shows that setting the detector efficiency for the background to one 