Triangle-free extremizers for warm-started QAOA lower bounds

Let $\textbf{LB}_{\kappa}(\theta)$ and $\textbf{LB}'_{\kappa}(\theta)$ be the stated inner-minimization problems, with variables $\mathbf r$, $\mathbf s$, $r_1$, $r_6$, $s$, $t$, $s_{1,2}$, and parameter $\kappa$. Triangle-free extremizer conjecture. For all initialization angles $\theta$, the optimal values in these minimization problems satisfy

$$\mathbf r=\mathbf s=\mathbf 0,$$

so that $G$ is triangle-free, and

$$r_1+r_6=1-\kappa.$$

Together, these conditions imply that the constraint

$$s+t+s_{1,2}+r_1+r_6\leq 1-\kappa$$

is tight. The source presents this as an unproved conjecture based on observed optimal values; its resolution is not given.