Inverse Turán conjecture for the four-cycle

Let $C_4$ denote the cycle of length four, and let $\operatorname{ex}^{-1}(k,H)$ denote the inverse Turán number of $H$, namely the maximum number of edges in a graph whose every $H$-free subgraph has fewer than $k$ edges. Inverse Turán conjecture for the four-cycle.

$$\operatorname{ex}^{-1}(k,C_4)=\frac{2\sqrt{2}}{3\sqrt{3}}k^{3/2}+o(k^{3/2}).$$

The paper gives upper and lower bounds for $\operatorname{ex}^{-1}(k,C_4)$ and states that this conjectured value is asymptotically sharp for the lower bound; the conjecture remains open in the supplied text.