Optimal girth-dependent bounds for the Bethe approximation of the permanent

For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison $$\mathrm{Bethe}(A)\le \mathrm{per}(A)\le 2^{n\mathrm{/}2}\mathrm{Bethe}(A)\mathrm{.}$$ The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g\ge 4$, then $$\mathrm{Bethe}(A)\le \mathrm{per}(A)\le 2^{2n\mathrm{/}g}\mathrm{Bethe}(A)\mathrm{.}$$ The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.