Three-terminal packing problem for the query q2

Determine whether there exists a linear function f such that, if more than f(ℓ) edges must be removed from a directed graph G to make the query q₂ false, then G contains ℓ pairwise edge-disjoint trees, each containing one vertex from each of S_G(a), S_G^-(b), and S_G^-(c), with no two trees sharing a vertex from the union of these three sets.

Background

The query q₂ asks for a recursively defined branching pattern in a directed graph, with all relevant paths required to have positive length. The authors formulate a concrete packing statement that would yield a constant-query tester for falsity of q₂. They also prove that any admissible function must satisfy f(ℓ) ≥ 2ℓ, so the unresolved issue is whether some linear function with at least this growth suffices.