Erdős–Sós order-form conjecture for bipartite host graphs

Every tree is bipartite. Write $T_{a,b}=(A,B)$ when the two parts of $T$ satisfy

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

For a bipartite graph $H$, let $\operatorname{ex}_{\mathrm{bip}}(N;H)$ be the maximum number of edges in an $H$-free bipartite graph on $N$ vertices. Order-form conjecture. For every tree $T_{a,b}$,

$$\operatorname{ex}_{\mathrm{bip}}(N;T_{a,b})\leq\begin{cases} \frac12(a-1)N& \text{if}\ a\geq2b-1,\\ (b-1)N& \text{if}\ a\leq2b-1. \end{cases}$$

The claim is motivated by two natural lower-bound constructions, whose comparison suggests a transition at $a=2b-1$. Its resolution is not supplied in the source material.