Factorization conjecture for weighted Aztec-region matching generating functions

Let $x,y,z$ be indeterminate edge weights, and let $A^{(i)}_{a,b,c}(x,y,z)$ and $F^{(i)}_{a,b,c}(x,y,z)$, for $i=1,2,3$, be the corresponding weighted versions of the graphs $A^{(i)}_{a,b,c}$ and $F^{(i)}_{a,b,c}$. Define

$$\alpha(a,b,c;x,y,z)$$

and

$$\beta(a,b,c;x,y,z)$$

by the piecewise formulas given in the source. Factorization conjecture. The matching generating functions of the weighted graphs $A^{(i)}_{a,b,c}(x,y,z)$ all have the form

$$\alpha(a,b,c;x,y,z)2^{X}(x^2+2xyz+2y^2z^2)^{Y}(2x^2+5xyz+4y^2z^2)^{Z}x^{T}y^{Q}z^{K},$$

for some $X,Y,Z,T,Q,K$ depending only on $a,b,c$. Similarly, the matching generating functions of the weighted graphs $F^{(i)}_{a,b,c}(x,y,z)$ all have the form

$$\beta(a,b,c;x,y,z)2^{X'}(x^2+2xyz+2y^2z^2)^{Y'}(2x^2+5xyz+4y^2z^2)^{Z'}x^{T'}y^{Q'}z^{K'},$$

for some $X',Y',Z',T',Q',K'$ depending only on $a,b,c$. The proposed factorization captures the observed structure of these matching generating functions; the supplied text gives no resolution.