Positive-isolated-part conjecture for clique-density optimizers

For $2\leq i\leq k$, let $a_i$ be coefficients and define

$$\lambda(G):=\sum_{i=2}^k a_i p(K_i,G),$$

where $p(K_i,G)$ is the induced density of $K_i$. Let $OPT$ denote the set of optimal vectors, and write $x_0$ for the coordinate corresponding to the isolated part. Positive-isolated-part conjecture. If $OPT$ contains $\mathbold{x}$ with $x_0>0$, then necessarily $x_0=1$. The paper presents this as an open question alongside other structural questions about optimal vectors; no resolution is supplied in the given text.