Exact bipartite Komlós–Sárközy–Szemerédi conjecture

A bipartite graph $G=(A,B,E)$ is balanced when its two parts have equal size, and $\delta(G)$ denotes its minimum degree. A balanced rooted tree on $2n$ vertices has $n$ vertices in each partition class, and $\Delta(T)$ denotes its maximum degree.

Exact bipartite Komlós–Sárközy–Szemerédi conjecture. For each $\gamma>0$, there are $c,n_0>0$ such that for every $n\ge n_0$, if $G=(A,B,E)$ is a balanced bipartite graph on $2n$ vertices with $\delta(G)\ge(\frac12+\gamma)n$, and $T$ is a balanced rooted tree on $2n$ vertices with $\Delta(T)\le\frac{cn}{\log n}$, then $T$ embeds in $G$ with the root of $T$ embedded in $A$.

This is proposed as the direct bipartite analogue of the Komlós–Sárközy–Szemerédi theorem. The source describes it as a belief rather than a proved result, so it remains open.