Plane paths in simple drawings of complete graphs

Simple drawings are drawings of graphs in the plane
such that vertices are distinct points in the plane,
edges are Jordan arcs connecting their endpoints, and
edges intersect at most once either in a proper crossing
or in a shared endpoint.
It is conjectured that every simple drawing of the
complete graph with $n$ vertices, $K_n$, contains a plane
Hamiltonian cycle, and consequently a plane Hamil-
tonian path. However, to the best of our knowledge,
$\Omega((\log n)^{1/6})$ is currently the best known lower bound
for the length of a plane path contained in any simple
drawing of $K_n$. We improve this bound to $\Omega(\log n / (\log \log n) )$.