Forest-Wilf equivalence implies Wilf equivalence

Let $S$ and $S'$ be sets of patterns. Forest-Wilf equivalence means that the two pattern sets have equal avoidance counts in rooted forests, while Wilf equivalence means equality of their avoidance counts in permutations.

Forest-Wilf implication conjecture. If $S$ and $S'$ are forest-Wilf equivalent, then $S$ and $S'$ are Wilf equivalent with respect to pattern avoidance in permutations.

This conjecture proposes that forest-Wilf equivalence is at least as strong as ordinary Wilf equivalence. The source gives no resolution; it immediately notes a stronger conjecture involving forest-structure-Wilf equivalence.