Connected forbidden graphs for all rational generalized Turán exponents

Determine whether, for every rational number \(\alpha\ge 1\), there exist connected graphs \(H_\alpha\) and \(F_\alpha\) such that \(\operatorname{ex}(n,H_\alpha,F_\alpha)=\Theta(n^\alpha)\).

Background

The paper proves that every rational exponent $\alpha\ge1$ can be realized as $$\operatorname{ex}(n,H_\alpha,F_\alpha)=\Theta(n^\alpha)$$, with $H_\alpha$ connected and of diameter at most 3. However, the compression construction used to replace a finite forbidden family by a single forbidden graph produces a forbidden graph $F_\alpha$ that is generally disconnected: its components encode the members of the finite forbidden family. The authors explain that the bridge in the construction prevents a connected component of a forbidden copy from using vertices on both sides, but does not make the resulting forbidden graph connected. They therefore explicitly ask whether both the counting graph and the single forbidden graph can always be required to be connected.