Self-reciprocity conjecture for rectangular tatami tiling generating functions

For a fixed number of rows $r\geq 1$, let $T_r(z)=f(z)/g(z)$, where $f(z)$ and $g(z)$ are relatively prime polynomials and $\deg(g(z))=n$.

Self-reciprocity conjecture. The denominator satisfies

$$g(z)=\left\{\begin{aligned} -z^n g\left(\frac{1}{z}\right),&\quad r\equiv 0\pmod{4},\\ -z^n g\left(-\frac{1}{z}\right),&\quad r\equiv 1\pmod{4},\\ z^n g\left(\frac{1}{z}\right),&\quad r\equiv 2\pmod{4},\\ z^n g\left(-\frac{1}{z}\right),&\quad r\equiv 3\pmod{4}. \end{aligned}\right.$$

This conjecture summarizes a mod-$4$ pattern observed in denominator coefficients; in particular, for $r\equiv 2\pmod{4}$ it implies that $g(z)$ is self-reciprocal. The source gives no resolution.