  In this paper, we prove an upper bound on the second non-zero Laplacian
eigenvalue on $n$-dimensional real projective space. The sharp result for
2-dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when
the metric degenerates to that of the disjoint union of a round projective
space and a sphere. That conjecture is open in higher dimensions, but this
paper proves it up to a constant factor that tends to 1 as the dimension tends
to infinity. Also, we introduce a topological argument that deals with the
orthogonality conditions in a single step proof.
