Nullity, determinant, and FAMEDness conjectures for ordered geometric triangulations

Let $T$ be an ordered and geometric triangulation of a knot complement. Let $\mathcal{A}$ and $\mathbf{B}$ be the matrices associated with $T$, and let the nullity of a matrix mean the codimension of its image.

Nullity, determinant, and FAMEDness conjecture. The following assertions hold:

1. The nullity of $\mathbf{B}$ is twice the nullity of $\mathcal{A}$.
2. If $\mathcal{A}$ and $\mathbf{B}$ are invertible, then

$$\det(\mathcal{A})=\pm 1,\qquad \det(\mathbf{B})=\pm 2.$$

3. If $\mathcal{A}$ and $\mathbf{B}$ are invertible, then $T$ is FAMED.

In particular, conditions (2), (3), and (4) of the FAMED condition are equivalent. These patterns were observed experimentally across hundreds of thousands of ordered triangulations, but the statement is presented as a conjecture because no general proof is supplied.