Erdős–Sós fixed-part conjecture for bipartite host graphs

Let $T_{a,b}=(A,B)$ be a tree whose bipartition satisfies

$$|A|=a,\qquad |B|=b,\qquad a\geq b.$$

For a bipartite graph $H$, let $\operatorname{ex}_{\mathrm{bip}}(m,n;H)$ be the maximum number of edges in an $H$-free bipartite graph with part-sizes $m$ and $n$. Fixed-part conjecture. For every tree $T_{a,b}$ and all $m\geq b-1$, $n\geq a-1$,

$$\operatorname{ex}_{\mathrm{bip}}(m,n;T_{a,b})\leq(b-1)n+\begin{cases} (a-b)m&\text{if}\ a\geq2b-1,\\ (b-1)m&\text{if}\ a\leq2b-1. \end{cases}$$

This refines the order-form problem by prescribing both part-sizes. The source motivates it through lower-bound constructions, but provides no resolution status for the conjecture.