Trinomial family conjecture for finding polynomials satisfying Property condition

Let $p$ be the prime associated with the finite field under consideration, and let $\mathcal{T}_{(n,k,a,b)}=\{x^n+ax^k+b\mid n>k>0;\ a,b\in\mathbb{Z}^*\}$ be the family of trinomials. Define

$$\mathcal{F}=\{\mathcal{T}_{(2i,k,a,b)}\mid 1\leq i,k,a,b\leq\log^2 p\}.$$

Trinomial family conjecture. The family $\mathcal{F}$ contains at least one polynomial satisfying Property condition for $r=2$.

This conjecture proposes an explicit bounded-parameter family for constructing a polynomial with the required property; the source leaves both a proof and a refutation open.