We address the problem of the fulfillment of the conjecture proposed by Jockers et al. (JKLMR conjecture) on the equality of the partition function of a supersymmetric gauged linear sigma model on the sphere $S^2$ and the exponential of the Kähler potential on the moduli space of Calabi-Yau manifolds. The problem is considered for a specific class of Calabi-Yau manifolds that does not belong to the Fermat type class. We show that the JKLMR conjecture holds when a Calabi-Yau manifold $X(1)$ of such type has a mirror partner $Y(1)$ in a weighted projective space that also admits a Calabi-Yau manifold of Fermat type $Y(2)$. Moreover, the mirror $X(2)$ for $Y(2)$ has the same special geometry on the moduli space of complex structures as the original $X(1)$.