  An ortho-polygon visibility representation $\Gamma$ of a $1$-plane graph $G$
(OPVR of $G$) is an embedding preserving drawing that maps each vertex of $G$
to a distinct orthogonal polygon and each edge of $G$ to a vertical or
horizontal visibility between its end-vertices. The representation $\Gamma$ has
vertex complexity $k$ if every polygon of $\Gamma$ has at most $k$ reflex
corners. It is known that $3$-connected $1$-plane graphs admit an OPVR with
vertex complexity at most twelve, while vertex complexity at least two may be
required in some cases. In this paper, we reduce this gap by showing that
vertex complexity five is always sufficient, while vertex complexity four may
be required in some cases. These results are based on the study of the
combinatorial properties of the B-, T-, and W-configurations in $3$-connected
$1$-plane graphs. An implication of the upper bound is the existence of a
$\tilde{O}(n^\frac{10}{7})$-time drawing algorithm that computes an OPVR of an
$n$-vertex $3$-connected $1$-plane graph on an integer grid of size $O(n)
\times O(n)$ and with vertex complexity at most five.
