The quarter-turn generating-polynomial factorization conjecture

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric alternating sign matrices. Quarter-turn polynomial factorization conjecture. There exists a sequence of polynomials $w_0(x),w_1(x),\ldots$ such that

$$Q_{2n+1}(x,1)=w_n(x)w_{n+1}(x)$$

if $n$ is even, and

$$Q_{2n-1}(x,1)=xw_n(x)w_{n+1}(x)$$

if $n$ is odd. Moreover,

$$Q_{4n}(x,1)=v_n(x)w_{2n}(x)$$

for suitable polynomials $v_n(x)$. The source reports the first factorization for $n=0,\ldots,7$ and the second for $n=1,\ldots,4$, with initial $v$'s and $w$'s tabulated; no general proof is given.