Palindromicity conjecture for twisted Neumann–Zagier determinant polynomials

Let $\mathbf{A}(t)$ and $\mathbf{B}(t)$ be the twisted Neumann–Zagier matrices of an ideal triangulation, with entries Laurent polynomials in $t$. A Laurent polynomial $p(t)$ is palindromic if

$$p(t)=\varepsilon t^r p(1/t)$$

for some $\varepsilon=\pm1$ and integer $r$.

Palindromicity conjecture. The Laurent polynomials $\det(\mathbf{A}(t))$ and $\det(\mathbf{B}(t))$ are palindromic.

This is a conjectural property of the twisted Neumann–Zagier matrices that the authors report checking in numerous examples; no resolution is supplied here.