The far-from-Turán positive discrepancy conjecture

For a graph $H$ on $n$ vertices, call $G$ $\varepsilon$-far from $H$ if at least $\varepsilon n^2$ edge changes are required to obtain a graph isomorphic to $H$. A Turán graph is a complete multipartite graph whose parts are as equal in size as possible.

Far-from-Turán discrepancy conjecture. For every $\varepsilon>0$, there exists $c>0$ such that, if $G$ is an $n$-vertex graph that is $\varepsilon$-far from every Turán graph, including the empty graph, then

$$\operatorname{disc}^{+}(G)\geq cn^{5/4}.$$

The conjecture proposes that graphs far from all Turán graphs have substantially larger positive discrepancy than the known $O(n)$ examples. The source gives a de Caen construction showing the $n^{5/4}$ scale is motivated, but the assertion remains open.