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We study the new case of the application of the JKLMR conjecture on theconnection between the exact partition functions of $\mathcal{N}=(2,2)$supersymmetric gauged linear sigma models (GLSM) on $S^2$ and special K\"ahlergeometry on the moduli spaces of Calabi-Yau manifold $Y$. The last ones ariseas manifolds of the supersymmetric vacua of the GLSM. We establish thiscorrespondence using the Mirror symmetry in Batyrev's approach. Namely,starting from the two-moduli non-Fermat Calabi-Yau manifold $X$ we constructthe dual GLSM with the supersymmetric vacua $Y$, which is the mirror for $X$.Knowing the special geometry on the complex moduli space of $X$ we verify themirror version of the JKLMR conjecture by explicit computation.

Partition functions of $\mathcal{N}=(2,2)$ supersymmetric sigma models and Special geometry for the two-moduli non-Fermat Calabi-Yau manifold