Quadratic denominator-degree conjecture for rectangular tatami tiling generating functions

For a fixed number of rows $r\geq 1$, let $T_r(z)$ be the ordinary generating function for tatami tilings of an $r\times n$ rectangle, and let $g(z)$ be its denominator. Write $r=4m+s$ with $s\in\{0,1,2,3\}$.

Denominator-degree conjecture. The degree of the denominator is

$$\deg(g(z))=\left\{\begin{aligned} 8m^2+2m+1,&\quad r\equiv 0\pmod{4},\\ 8m^2+4m+2,&\quad r\equiv 1\pmod{4},\\ 8m^2+10m+4,&\quad r\equiv 2\pmod{4},\\ 8m^2+8m+6,&\quad r\equiv 3\pmod{4}. \end{aligned}\right.$$

The formulas extend the mod-$4$ pattern observed in computed denominator degrees, but the source gives no resolution.