Forest-Wilf equivalences for pairs of patterns

A forest-Wilf equivalence means equality of the relevant avoidance counts for rooted forests. The notation $S\sim S'$ denotes that the pattern sets $S$ and $S'$ are forest-Wilf equivalent.

Forest-Wilf equivalence conjecture. The following three equivalences hold:

$$\{123,2413\}\sim\{132,2314\},$$ $$\{123,3142\}\sim\{132,3124\},$$ $$\{213,4123\}\sim\{213,4132\}.$$

These are proposed as further forest-Wilf equivalences; the source gives no resolution or supporting theorem beyond presenting them as conjectures.