  In a paper by Bella, Tokg\"os and Zdomskyy it is asked whether there exists a
Tychonoff space $X$ such that the remainder of $C_p(X)$ in some
compactification is Menger but not $\sigma$-compact. In this paper we prove
that it is consistent that such space exists and in particular its existence
follows from the existence of a Menger ultrafilter.
