Generalized mirror-transformation conjecture for projective hypersurfaces

Let $P_m$ denote the set of partitions of $m$, let $l(\sigma_m)$ be the length of a partition, and let $C_m^d(\sigma_m)$ be the coefficients associated with partitions in the proposed generalized mirror transformation. Let $L_n^{N,k,d}$ be the true structure constants and $\tilde{L}_n^{N,k,d}$ the virtual structure constants. For each partition $\sigma_m\in P_m$, let $G_{d-m}^{N,k,d}(n;\sigma_m)$ be a degree $d-m$ weighted homogeneous polynomial in the virtual structure constants. Generalized mirror-transformation conjecture. The map $m_N^k$ is given by

$$L_n^{N,k,d}=\sum_{m=0}^{d-1}\sum_{\sigma_m\in P_m}\left(C_m^d(\sigma_m)\cdot\prod_{i=1}^{l(\sigma_m)}\left(\tilde{L}_{1+(k-N)d_i}^{N,k,d_i}\right)\right)\cdot G_{d-m}^{N,k,d}(n;\sigma_m),$$

where $G_{d-m}^{N,k,d}(n;\sigma_m)$ is a degree $d-m$ weighted homogeneous polynomial of $\tilde{L}_n^{N,k,d}$. This is proposed as a generalization of the mirror transformation from Calabi–Yau hypersurfaces; the paper gives structural evidence and partial formulas but does not prove the general statement.