Mirror correspondence for class TG \Q-Fano threefolds

A qG-deformation family is a family of \Q-Fano threefolds up to \Q-Gorenstein deformation. A \Q-Fano threefold $X$ is of class TG if it occurs as the general fiber of a qG-degeneration with reduced fibers and special fiber a normal toric variety. Laurent polynomials in three variables are considered up to mutation-equivalence; a Laurent polynomial is rigid maximally mutable if it has the rigidity and maximal mutation properties specified in the mirror-symmetry framework.

Mirror correspondence. There is a bijective correspondence between qG-deformation families of \Q-Fano threefolds $X$ of class TG and mutation-equivalence classes of rigid maximally mutable Laurent polynomials $f$ in three variables. Under this correspondence, the regularized quantum period $\widehat{G}_X$ coincides with the classical period $\pi_f$.

This is the proposed mirror-symmetry classification of class TG \Q-Fano threefolds, relating deformation families to mutation classes of Laurent polynomials and matching their quantum and classical periods. The supplied text does not state whether the correspondence is proved or remains conjectural.