LG mirror symmetry conjecture for BHK mirror pairs

Let $W:\mathbb{C}^N\to\mathbb{C}$ be an invertible quasihomogeneous polynomial with an isolated critical point at the origin, written as

$$W=\sum_{i=1}^{N}\prod_{j=1}^{N}x_j^{a_{ij}},$$

and let $G_W$ be its maximal group of diagonal symmetries. Define the transpose polynomial by

$$W^T=\sum_{i=1}^{N}\prod_{j=1}^{N}x_j^{a_{ji}}.$$

LG mirror symmetry conjecture. Up to a change of variables, the generating function of the FJRW theory at all genera for $(W,G_W)$ can be identified with the generating function of the Saito-Givental theory of $W^T$. This conjecture asserts the equivalence of the mathematical Landau-Ginzburg A- and B-models for BHK mirror pairs, but its general status is not established.