The regular Turán bound for 3-chromatic graphs built from bipartite graphs

Let $F$ be a 3-chromatic graph obtained from a bipartite graph by adding a set of independent edges into one part. Write $\mathrm{rex}(n,F)$ for the maximum number of edges in an $n$-vertex, regular, $F$-free graph.

Regular Turán bound. For such a graph $F$,

$$\mathrm{rex}(n,F)\le (1+o(1))\frac{n^2}{5}.$$

This bound is presented in the source as an assertion for a class of 3-chromatic graphs in the study of regular Turán numbers. Because the statement appears as a standalone claim but is not identified there as a conjecture or accompanied by a resolution status, its status should be checked against the surrounding paper; it is recorded here as open.