  The Reidemeister number $R(\varphi)$ of a group automorphism $\varphi \in
\mathrm{Aut}(G)$ encodes the number of orbits of the $\varphi$-twisted
conjugation action of $G$ on itself, and the Reidemeister spectrum of $G$ is
defined as the set of Reidemeister numbers of all of its automorphisms. We
obtain a sufficient criterion for some groups of triangular matrices over
integral domains to have property $R_\infty$, which means that their
Reidemeister spectrum equals $\{\infty\}$. Using this criterion, we show that
Reidemeister numbers for certain soluble $S$-arithmetic groups behave
differently from their linear algebraic counterparts -- contrasting with
results of Steinberg, Bhunia, and Bose.
