Weighted uniform-edge conjecture for random bipartite matchings

Let $A=(a_{ij})$ be a doubly stochastic $n\times n$ matrix, and let $W=(w_{ij})$ be a matrix of edge weights satisfying

$$\sum_{i,j=1}^{n}w_{ij}a_{ij}=1.$$

Let $R$ be the random bipartite graph in which edge $(i,j)$ appears independently with probability $a_{ij}$. Weighted uniform-edge conjecture. The expected weight of a maximum-weight matching in $R$ is minimized when $a_{ij}=w_{ij}=1/n$ for every $(i,j)$. This weighted form is motivated by the duality between contention-resolution schemes and expected matching quantities; proving it would yield an optimal $\gamma$-balanced contention-resolution scheme for bipartite matchings.