The half-turn-symmetric alternating sign matrix factorization conjecture

Let $H_n(x,y)$ be the weight-generating function for half-turn-invariant $n$ by $n$ alternating sign matrices, and let $Z_n(x,y,0)$ and $Z_{n-1}(x,y,1)$ be the corresponding plane-partition generating functions defined in the source. Half-turn factorization conjecture. It is conjectured that

$$H_{2n}(x,y)=Z_n(x,y,0)Z_{n-1}(x,y,1).$$

The proposed factorization suggests a bijection between half-turn-symmetric $2n$ by $2n$ alternating sign matrices and the Cartesian product of $n$ by $n$ alternating sign matrices with cyclically symmetric plane partitions in the box ${\cal X}_n$. The formula is described as observed, and the source gives no proof or resolution.