The complement-closed Lubell bound conjecture for bipartite graphs

Let $G$ be a bipartite graph that is not a matching. For a poset $P$, let $\mathop{}\!\mathrm{La}_{sym}(n,P)$ denote the maximum size of a $P$-free complement-closed family of subsets of $[n]$, and let $P_{G,A}$ be the poset associated with the part $A$ of $G$. The complement-closed Lubell bound conjecture.

$$\mathop{}\!\mathrm{vex}(n,G)=2^{n-1}+\frac{1}{2}\mathop{}\!\mathrm{La}_{sym}(n,P_{G,A})+o\left(\binom{n}{n/2}\right).$$

This conjecture asserts that the complement-closed poset-free-family upper bound gives the correct asymptotics for every bipartite graph that is not a matching. The source provides no resolution, so the conjecture remains open.