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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 $\log\log(g)\leq\log\epsilon +\log\log n$?

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