Shape-Wilf-equivalence is not closed under inversion

We show that shape-Wilf-equivalence is not closed under inversion: of the seven inverse pairs of length-four patterns, exactly one is shape-Wilf-equivalent, and it is precisely the pair generated by the known equivalences of Backelin-West-Xin and Stankova-West. Since inverse patterns are always Wilf-equivalent and lifting commutes with inversion, every inverse pair satisfies the hypothesis of a question of Burstein (Permutation Patterns 2025; Question 3.6 of Vatter's problem collection, arXiv:2602.16355): if the lifts of two patterns are Wilf-equivalent, must the patterns be shape-Wilf-equivalent? Each of the six separated pairs is a counterexample, the simplest witnessed by an explicit 34-cell Ferrers board on which 2341 and 4123 have 392 and 394 avoiding full placements. It follows that shape-Wilf-equivalence, while closed under right direct sums, is not closed under left direct sums. All counts were verified by four independent implementations; machine-readable certificates and all verification scripts are included in the supplementary archive.