Gurvits's exponential BP upper-bound conjecture for the permanent

Let $p$ be a non-negative matrix, and let $f(n)$ be the factor in the BP upper bound

$$\operatorname{perm}(p)\leq Z_{o\text{-}BP}(p)f(n).$$

Gurvits's BP upper-bound conjecture. For any non-negative $p$, $f(n)$ is asymptotic to $\sqrt{2}^n$. This conjecture concerns the worst-case multiplicative gap between the permanent and the BP estimate; the earlier claim that $f(n)$ is asymptotic to $\sqrt n$ was disproved by an explicit counterexample. The source gives no resolution of the exponential-factor conjecture.