Compatible translates from independent pools

For every positive integer $D$ there is an integer $K\ge 6$ such that for every $\varepsilon>0$ there is a threshold $Q$ with the following property. Let $F$ be a finite field with $q=|F|\ge Q$ and let $n\le Dq$. For $1\le i\le n$ let $W_i\subseteq F^2$ and let $D_i$ be a set of directions with $|W_i|,|D_i|\le D$, and assume each direction belongs to $D_i$ for at most $D$ indices $i$. For distinct $i,j$, translates $b+W_i$ and $b'+W_j$ conflict if there are $u\in W_i$, $v\in W_j$ and $d\in D_i\cup D_j$ with

Draw $K$ independent uniform shifts $b_{i,1},\dots,b_{i,K}\in F^2$ for each $i$, independently for all indices. Then with probability at least $1-\varepsilon$ there is a choice of indices $k_i\in\{1,\dots,K\}$ for which the translates $b_{i,k_i}+W_i$ have no conflicts.

This is Theorem 4.1 of the paper, proved through an independent-transversal argument in the tradition of Haxell. It separates the independent random choices needed by the deletion estimates from the geometric compatibility needed for exceptional edges to intersect.