Determine the worst-case connected bipartite Turán ratios

Determine exactly, or improve the known lower bounds for, the worst-case asymptotic ratios \(\gamma_{b,c}=\liminf_{k\to\infty}\inf_{T\in\mathcal T_k}\limsup_{n\to\infty}\frac{\operatorname{ex}_{b,c}(n,T)}{(|T|-2)n}\) and \(\gamma_{b\ne c}=\liminf_{k\to\infty}\inf_{T\in\mathcal T_k}\limsup_{n\to\infty}\frac{\operatorname{ex}_{b,c}(n,T)}{\operatorname{ex}_{b}(n,T)}\).

Background

The paper introduces asymptotic ratios measuring how small connected bipartite Turán numbers can be relative to the normalization $(|T|-2)n$ and to the corresponding unrestricted bipartite Turán numbers. It proves $\gamma_b=2/3$ and derives lower bounds for the connected ratios, including numerical lower bounds for $\gamma_{b,c}$ and $\gamma_{b\ne c}$. The exact values of these two parameters, or stronger lower bounds, remain unresolved.