The unrefined-limit identity for the refined index

Let $g^{\rm (ref)}_n(\{\check\gamma_i,c_i\},y)$ be defined by the refined generating-function construction and the reference solution $F^{\rm (ref)}_n$, with coefficients $b_n$ fixed by the Bernoulli-number formula. Let $g^{(0)}_n(\{\check\gamma_i,c_i\})$ be the unrefined function. Unrefined-limit conjecture.

$$\lim_{y\to1}g^{\rm (ref)}_n(\{\check\gamma_i,c_i\},y)=g^{(0)}_n(\{\check\gamma_i,c_i\}).$$

This identifies the refined construction with the unrefined index in the limit $y\to1$; the supplied context establishes smoothness of the limit but gives no resolution of the identity itself.