The odd half-turn-symmetric alternating sign matrix factorization conjecture

Let $H_n(x,y)$ be the generating function for half-turn-symmetric alternating sign matrices, and let $R_n(x,\mu)$, $T_n(x,\mu)$, and the polynomials $S_{4n+1}(x),S_{4n-1}(x)$ be as defined in the source. Odd half-turn factorization conjecture. It is conjectured that

$$H_{4n+1}(x,1)=R_n(x,0)T_n(x,1)S_{4n+1}(x),$$

and

$$H_{4n-1}(x,1)=R_{n-1}(x,1)T_n(x,0)S_{4n-1}(x),$$

where $S_1(x),S_3(x),\ldots$ are certain polynomials. The source reports this factorization as observed and tabulates initial examples, but gives no general proof.