The Baumslag-Gersten group and a problem of Olshanskii

We prove that a certain representation of the Baumslag-Gersten one-relator group $\mathrm{B}\mathrm{G}(1,2)$ by germs of continuous functions is not faithful. This gives a negative answer to a problem of A. Yu. Olshanskii from 2010 (Problem 17.99 in the Kourovka Notebook).