(K_{1,2,2,2}\) has no \(n\)-fold planar cover graph for \(n<14\)

S. Negami conjectured in \(1988\) that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It follows from the works of D. Archdeacon, M. Fellows, P. Hliněn\'{y}, and S. Negami that this conjecture is true if the graph \(K_{1, 2, 2, 2}\) has no finite planar cover. We prove a number of structural results about putative finite planar covers of \(K_{1,2,2,2}\) that may be of independent interest. We then apply these results to prove that \(K_{1, 2, 2, 2}\) has no planar cover of fold number less than \(14\).