The far-from-Turán second-eigenvalue 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. Let $\lambda_2$ denote the second largest eigenvalue of the adjacency matrix of a regular graph.

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

$$\lambda_2\geq cn^{1/4}.$$

This is the analogous spectral problem to the preceding discrepancy conjecture. The source presents it as an open question motivated by constructions far from Turán graphs with second eigenvalue of order $n^{1/4}$.