The quarter-turn-symmetric alternating sign matrix factorization conjecture

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric $n$ by $n$ alternating sign matrices, and let $H_n$ and $A_n$ denote the half-turn-symmetric and unrestricted alternating sign matrix generating functions defined in the source. Quarter-turn factorization conjecture. For $n\geq 1$, it is conjectured that

$$Q_{4n}(1,y)=yH_{2n}(1,y,0)A_n(1,y)^2,$$ $$Q_{4n+1}(1,y)=yH_{2n+1}(1,y)A_n(1,y)^2,$$

and

$$Q_{4n-1}(1,y)=yH_{2n-1}(1,y)A_n(1,y)^2.$$

The formulas were verified in the source only for $n=1,2,3,4$ and suggest bijections with Cartesian products involving other alternating sign matrix classes and cyclically symmetric plane partitions.