A note on the string topology BV-algebra for $S^2$ with $Z_2$ coefficients

Luc Menichi showed that the BV algebras on $H^{\bullet }(L{S}^2;Z_2)[-2]$ coming from string topology and the one on $H{H}^{\bullet }(H^{\bullet }(S^2;Z_2),H^{\bullet }(S^2;Z_2))$ using Poincaré duality on $H^{\bullet }(S^2;Z_2)$ are not isomorphic. In this note we show how one can obtain the string topology BV algebra on Hochschild cohomology using a Poincaré duality structure with higher homotopies. This Poincaré duality (with higher homotopies) on cohomology is induced by a local Poincaré duality (with higher homotopies) on the cochain level.