Conjecture on real structures in the LG/CY correspondence

Let $R_f$ be the Milnor ring, let $\Delta_f$ be the Landau–Ginzburg Laplacian, and let $r:\bigoplus_{a=0}^{\infty}R_f^{(n+2)a}\to H^n_{\operatorname{prim}}(X_f)$ be given by

$$r([A])=({\rm res}\,\Omega_A)^{n-a,a},\qquad \forall\,\operatorname{deg}A=(n+2)a.$$

Suppose $[A],[B]\in R_f$ and $(n-1)$-forms $\mu,\nu$ are such that $\alpha=A\,dz_1\wedge\cdots\wedge dz_n+\bar{\partial}_f\mu$ and $\beta=B\,dz_1\wedge\cdots\wedge dz_n+\bar{\partial}_f\nu$ are $\Delta_f$-harmonic forms. Conjecture on real structures. If $\alpha=\bar{\beta}$, then $r(A)=\overline{r(B)}$. This proposes that complex conjugation on the Landau–Ginzburg harmonic-form side corresponds, under $r$, to complex conjugation on the Calabi–Yau side; the source gives no evidence of resolution.