The Feedback-Exponent Conjecture for bounded minimum feedback vertex number

Let $k_0\geq 1$ be an integer. For a graph $H$, write $\operatorname{ex}(n,H)$ for its Turán number, and call a rational number $r\in[1,2]$ realizable as a Turán exponent if there is a graph $H$ such that

$$\operatorname{ex}(n,H)=\Theta(n^r).$$

The minimum feedback vertex number (FVN) of a graph is the minimum number of vertices whose removal makes the graph acyclic. Feedback-Exponent Conjecture. There exists a rational number $p/q\in[1,2]$ that cannot be realized as the Turán exponent of any bipartite graph whose minimum FVN is at most $k_0$.

This conjecture asks whether bounded minimum feedback vertex number restricts the rational exponents realizable by bipartite graphs. It is motivated by the Rational Exponent Conjecture of Erdős and Simonovits and by examples showing that, for bipartite graphs with minimum FVN $2$, the Turán exponent is not determined solely by the minimum cycle length; the status of the proposed restriction is not established in the supplied text.