  An orientation of $G$ is a digraph obtained from $G$ by replacing each edge
by exactly one of two possible arcs with the same endpoints. We call an
orientation \emph{proper} if neighbouring vertices have different in-degrees.
The proper orientation number of a graph $G$, denoted by $\vec{\chi}(G)$, is
the minimum maximum in-degree of a proper orientation of G. Araujo et al.
(Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant $c$
such that $\vec{\chi}(G)\leq c$ for every outerplanar graph $G$ and showed that
$\vec{\chi}(G)\leq 7$ for every cactus $G.$ We prove that $\vec{\chi}(G)\leq 3$
if $G$ is a triangle-free $2$-connected outerplanar graph and
$\vec{\chi}(G)\leq 4$ if $G$ is a triangle-free bridgeless outerplanar graph.
