The determinantal basis criterion for extending generalized spline modules

Let $(G,\beta)$ be an edge-labeled graph over a GCD domain $R$, and let $F_1,F_2,\ldots,F_n\in\hat{R}_G$ be splines. The determinant of these splines is denoted by $\lvert F_1,F_2,\ldots,F_n\rvert$, and $\hat{Q}_G$ is the associated element of $R$. Determinantal basis criterion. If $\{F_1,F_2,\dots,F_n\}$ forms an $R$-module basis of $\hat{R}_G$, then

$$\lvert F_1,F_2,\ldots,F_n\rvert=u\hat{Q}_G$$

for some unit $u\in R$. This conjecture seeks the converse to the sufficient determinant criterion established earlier, and a general determinantal criterion for arbitrary graphs and rings remains open.