Weighted rainbow clique conjecture

Let $r\ge 3$, $k\ge\binom{r}{2}$, and let $G_0\cong K_n$ be a complete graph on $[n]$. A $k$-weighted graph is a pair $\mathcal{G}=(G_0,f)$ with an edge-weighting $f$. A weight sequence bound $(1,2,\ldots,\binom{r}{2})$ means that every $K_r$ has its nondecreasing edge-weight sequence lexicographically at least this sequence. Define

$$k_2=\left\lceil\frac{\left(\binom{r}{2}-1\right)\binom{n}{2}}{t_{r-1}(n)}\right\rceil,\qquad k_1=k_2-1.$$

Weighted rainbow clique conjecture. If $n\ge r$, $G_0$ contains no $K_r$ with weight sequence bound $(1,2,\ldots,\binom{r}{2})$, then: (i) if $k=k_1$, $f(G_0)\le(\binom{r}{2}-1)\binom{n}{2}$; and (ii) if $k=k_2$, $f(G_0)\le k\,t_{r-1}(n)$.

This weighted statement is introduced as a conjectural reformulation implying the rainbow Turán clique bound. The source proves it for $r=4,5$; the general case remains open.