Gurvits's doubly stochastic permanent inequality conjecture

Let $\phi$ be a doubly stochastic $n\times n$ matrix, so that its entries are non-negative and every row and column sums to $1$. The permanent is denoted by $\operatorname{perm}(\phi)$. Gurvits's permanent inequality conjecture. The following inequality holds:

$$\operatorname{perm}(\phi)\leq \sqrt{2}^n\prod_{(i,j)}(1-\phi_{ij})^{1-\phi_{ij}}.$$

If true, this inequality would imply a deterministic polynomial-time algorithm approximating the permanent of an $n\times n$ non-negative matrix within relative factor $\sqrt{2}^n$. The source does not state that the inequality has been proved or disproved.