Exact star-colored Turán bound conjecture

For integers $r$ and $s\geq r-1$, let $G_1,\ldots,G_s\subseteq\binom{[n]}{2}$ be graphs whose edges are $(r-1)$-star edge-colored and whose underlying graphs are $K_r$-free. Here $e(G_i)$ denotes the number of edges of $G_i$, and $t(n,r-1)$ denotes the Turán number of $K_r$ on $n$ vertices. Exact star-colored Turán bound conjecture.

$$\sum_{i=1}^{s}e(G_i)\leq s\cdot t(n,r-1).$$

The preceding theorem establishes the same bound only up to an additive term $sn$ in the regime $s\geq r-1$; removing that error term would give the exact extremal value in the corresponding application. The source leaves this conjecture open.