  We study the new case of the application of the JKLMR conjecture on the
connection between the exact partition functions of $\mathcal{N}=(2,2)$
supersymmetric gauged linear sigma models (GLSM) on $S^2$ and special K\"ahler
geometry on the moduli spaces of Calabi-Yau manifold $Y$. The last ones arise
as manifolds of the supersymmetric vacua of the GLSM. We establish this
correspondence using the Mirror symmetry in Batyrev's approach. Namely,
starting from the two-moduli non-Fermat Calabi-Yau manifold $X$ we construct
the 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 the
mirror version of the JKLMR conjecture by explicit computation.
