Asymptotic sharpness conjecture for colored generalized Turán numbers

Let $G$ be a graph whose edges are colored by $1,\dots,k$, and let $G_i$ be the subgraph consisting of the edges of color $i$. For graphs $H_1,\dots,H_k$ and an $F$-free graph $G$ on $n$ vertices, define

$${\mathrm{ex}^{\mathrm{col}}}(n,(H_1,\dots,H_k),F)=\max_G\sum_{i=1}^k {\mathcal N}(H_i,G_i).$$

Here ${\mathrm{ex}}(n,H_i,F)$ denotes the usual generalized Turán number.

Colored asymptotic sharpness conjecture.

$${\mathrm{ex}^{\mathrm{col}}}(n,(H_1,\dots,H_k),F)=(1+o(1))\max_{i\le k}{\mathrm{ex}}(n,H_i,F).$$

The preceding observations show that the colored generalized Turán number has order of magnitude $\Theta(\max_{i\le k}{\mathrm{ex}}(n,H_i,F))$. The conjecture asserts that the corresponding lower bound is asymptotically sharp, but the supplied text does not indicate whether this is known in any particular cases.