Weighted Turán conjecture for two-dimensional vector weights

Let $\omega$ be sufficiently large and let $G=(V,E)$ be a graph with clique number $\omega$. Assign a vector $\vv_i\in\mathbb R^2$ to each vertex $i$, and define

$$\ww_{i,j}=\langle\vv_i,\vv_j\rangle^2$$

for each pair of vertices $(i,j)$. Suppose that $\langle\vv_i,\vv_j\rangle\geq0$ for every $\{i,j\}\in E$. Weighted Turán conjecture. Then

$$\sum_{\{i,j\}\in E}\ww_{i,j}\leq\frac12\left(1-\frac1\omega\right)\sum_{i,j\in V}\ww_{i,j},$$

where the sum on the left counts each edge once.

This is presented as a weighted analogue of Turán's theorem and as a reformulation of the preceding conjecture on $\chi_{\mathrm{vec},3}”(G)$; no resolution is given in the source.