# Biadjacency permanent upper bound in terms of genus of graph?

Take $$M$$ to be biadjacency of a planar balanced bipartite on $$2n$$ vertices with genus $$g$$.

Is it true for every $$\epsilon\in(0,1)$$ there is a $$c_\epsilon>0$$ such that $$\log\log(permanent(M))\leq\log c_\epsilon +\log n+\log\log n$$ at every $$g$$ with $$\log\log(g)\leq\log\epsilon +\log\log n$$?