  A 4-regular planar graph $G$ is said to be circle representable if there
exists a collection of circles drawn on the plane such that the touching and
crossing points correspond to the vertices of $G$, and the circular arcs
between those points correspond to the edges of $G$. Lov\'asz (1970)
conjectured that every 4-regular planar graph has a circle representation, but
an infinite family of counterexamples was given by Bekos and Raftopoulou
(2015). We reduce the order of the smallest known counterexamples among simple
graphs from 822 to 68 based on a multigraph counterexample of order 12.
