Pontiveros's conjecture on dense sets with incomplete sums of dilates

Let $p$ be a prime, let $k$ be a positive integer, and let $\lambda_1,\dots,\lambda_k$ be nonzero integers viewed in $\mathbb{F}_p$. For a set $A\subseteq\mathbb{F}_p$, write $\lambda A=\{\lambda a:a\in A\}$ and let $\varepsilon>0$. Pontiveros's conjecture. There exists $\delta=\delta(\lambda,\varepsilon)>0$ such that, for every sufficiently large prime $p$, there is a set $A\subseteq\mathbb{F}_p$ satisfying

$$|A|>(1/k-\varepsilon)p$$

and

$$|\lambda_1A+\dots+\lambda_kA|\leqslant(1-\delta)p.$$

The statement is refuted: the paper explicitly notes that its main theorem disproves it for $k=3$ and $(\lambda_1,\lambda_2,\lambda_3)=(1,1,-2)$, and gives a further explicit counterexample with six pairwise distinct coefficients.