  A set of vertices $W$ of a graph $G$ is a total $k$-dominating set when every
vertex of $G$ has at least $k$ neighbors in $W$. In a recent article, Chiarelli
et al.\ (Improved Algorithms for $k$-Domination and Total $k$-Domination in
Proper Interval Graphs, Lecture Notes in Comput.\ Sci.\ 10856, 290--302, 2018)
prove that a total $k$-dominating set can be computed in $O(n^{3k})$ time when
$G$ is a proper interval graph with $n$ vertices and $m$ edges. In this note we
reduce the time complexity to $O(m)$ for $k=2$.
