Determinant asymptotic conjecture for the Corner System

Let $F$ be a skew finite subset of $\boldsymbol{\backslashell}_1^N$, and let $r\in(0,\operatorname{skew}(F)/2)$. Define

$$k:=2^{N-1}N\cdot\#F,$$

so that $k$ is the number of $1$-dimensional faces in $\overline{C}_F(r)$. Determinant asymptotic conjecture. For every such $F$ and $r$,

$$\det\bigl(Z_{\overline{C}_F(r)^{(0)}}\bigr)=4^k r^k+O(r^{k+1}).$$

The conjecture arose from numerical experiments involving determinant expansions used to study the limits of coefficients in the Corner System. It is presented as a potentially useful pattern, but no proof or resolution is given in the source.