Commutative $BV_\infty$ algebras, their morphisms and $\frac{\infty}{2}$-variation of Hodge structures

We study morphisms between commutative $BV_\infty$ algebras and show that, under suitable additional assumptions, a quasi-isomorphism of commutative $BV_\infty$ algebras induces an identification of $\frac{\infty}{2}$-variations of Hodge structures with polarizations, and consequently of Frobenius manifolds. An explicit example arising from singularity theory is provided to illustrate the result.