Bipartite Graphs and Matchings · Hard problem

Men $1,2,3,4$ rank women cyclically: $1: A,B,C,D$; $\;2: B,C,D,A$; $\;3: C,D,A,B$; $\;4: D,A,B,C$. Women rank men cyclically too: $A: 2,3,4,1$; $\;B: 3,4,1,2$; $\;C: 4,1,2,3$; $\;D: 1,2,3,4$. Among all $24$ possible perfect matchings of these $4$ men to these $4$ women, how many are stable (have no blocking pair)?