Trees · Hard problem

Two $4$-cycles, $1\text{-}2\text{-}3\text{-}4\text{-}1$ and $5\text{-}6\text{-}7\text{-}8\text{-}5$, are joined by two separate connecting edges, $15$ and $26$ (vertices $1,\ldots,8$; edges $12,23,34,41,56,67,78,85,15,26$). How many spanning trees does this graph have?