Identification conjecture for the pre-quotient Calabi–Yau complete intersection

Let $X$ be the smooth Calabi–Yau complete intersection whose existence is conjectured to match the $A$-periods and GKZ system of $Y_{\mathrm{nc}}$, and let $X_{\mathrm{pre}}$ be the geometrically constructed Calabi–Yau complete intersection pre-quotient. Pre-quotient identification conjecture. The construction $X_{\mathrm{pre}}$ is conjectured to give the desired $X$, namely $X_{\mathrm{pre}}$ is smooth and

$$X_{\mathrm{pre}}=X.$$

This identifies the geometric pre-quotient with the complete intersection predicted by the preceding conjecture. The source provides geometric motivation and examples, but no proof of smoothness or equality in general.