Weighted-graph reformulation of the extremal partition conjecture

Fix integers $s$ and $t$ with $3\leq s\leq t-2$. A weighted graph admitting a $(b,t-1-b)$-partition has $b$ vertices and $t-1-b$ parts satisfying the defining partition conditions for the weighted-graph reduction. Weighted-graph reformulation. The maximum $K_s$-density among weighted graphs admitting a $(b,t-1-b)$-partition is attained when

$$b=\max\{s,\lfloor t/2\rfloor\}.$$

This is the weighted-graph form of the proposed extremal periodicity principle; the supplied text explains that the paper proves both conditions under which it holds and counterexamples, while no complete resolution is stated here.