Finite-support conjecture for clique-density optimizers

For $2\leq i\leq k$, let $a_i$ be coefficients and define a graph parameter by

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

where $p(K_i,G)$ is the induced density of the clique $K_i$. Let $OPT$ denote the set of optimal vectors, and let $\mathbold{x}$ be such a vector. Finite-support conjecture. For every $\lambda(G):=\sum_{i=2}^k a_i p(K_i,G)$, every $\mathbold{x}\in OPT$ has finite support. This is stated as an open question in the paper; the supplied context gives no resolution.