Contractible Subgraphs of Contraction Critically Quasi $5$-Connected Graphs

Let $G$ be a contraction critically quasi $5$-connected graph on at least $14$ vertices. If there is a vertex $x\in V_4(G)$ such that $G[N_G(x)]\cong K_{1,3}$ or $G[N_G(x)]\cong C_4$, then $G$ has a quasi $5$-contractible subgraph $H$ such that $0<\mathrm{\parallel }V(H)\mathrm{\parallel }<4$.