Regular Turán extremal conjecture for graphs with chromatic number at most k

Let $H$ be a graph with chromatic number $\chi(H)\le k$ and let $k\ge 3$. Write $T(n,k)$ for the $n$-vertex Turán graph with $k$ parts, and let $\mathcal N(H,G)$ denote the number of copies of $H$ in a graph $G$. The regular Turán number $\operatorname{rex}(n,H,K_{k+1})$ is the maximum number of copies of $H$ in an $n$-vertex regular $K_{k+1}$-free graph. Regular Turán extremal conjecture. Then

$$\operatorname{rex}(n,H,K_{k+1})=(1+o(1))\mathcal N(H,T(n,k)).$$

Moreover, if $n$ is sufficiently large and divisible by $k$, then

$$\operatorname{rex}(n,H,K_{k+1})=\mathcal N(H,T(n,k)).$$

The conjecture asks whether the known asymptotic and exact extremal behavior for ordinary Turán problems persists under the regularity restriction. The preceding discussion notes that known counterexamples to the corresponding unrestricted statement are far from regular; whether regular counterexamples exist is left open here.