Multiplicity of closed geodesics on odd-dimensional bumpy Finsler compact space form

Let $M=S^{2n+1}\mathrm{/}\mathrm{\Gamma }$ be a manifold diffeomorphic to a compact space form, endowed with a bumpy Finsler metric $F$. In this paper, we prove that there exist at least $n+3$ contractible minimal closed geodesics on $M$, assuming that all contractible minimal closed geodesics have non-zero Morse indices. Here, a contractible closed geodesic is called minimal if it is not an iteration of any other contractible closed geodesic. In particular, this confirms the generalized Anosov-Katok conjecture for the bumpy quotient manifold $(S^3\mathrm{/}\mathrm{\Gamma },F)$ under the non-zero Morse index assumption.