The determinant criterion for finiteness of rational points

Let $V(k)$) be the set of $k$-rational points of a projective variety $V$ over a number field $k$, and let $A^1_{\infty}$ be the infinite matrix associated with $V$ in the paper. Determinant criterion for rational points.

$$|V(k)|=\begin{cases} \text{finite}, & \text{if } \det(1-A^1_{\infty})=0,\\ \text{infinite}, & \text{otherwise.} \end{cases}$$

This conjecture proposes that the determinant of the matrix attached to $V$ distinguishes varieties with finitely many rational points from those with infinitely many. Its status is not established in the supplied material.