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Yes it does. Geometrically the BV operator is gotten by considering the parametrized moduli space of solutions to the Floer equation on the cylinder obtained by rotating one end of the cylinder. This is explained (rather briefly) in Section (8a) of Seidel's "Biased View of Symplectic Cohomology," see also Remark 5.7 of Bourgeois and Oancea's "The Gysin exact sequence for $S^1$-equivariant symplectic homology"