$K_{1,2,2,2}$ 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. Hliněný, 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 $14$.