Refined decomposition conjecture for alternating sign matrix enumerations

Let $A_n(z,\rho):=A_n(z,\rho,1)$, and for odd order define

$$B_{2m+1}(z,\rho)=\sum_{A\in\operatorname{HSASM}(2m+1)}z^{\frac{\mu(A)-m}{2}}\rho^{f(A)-2},$$

where $\operatorname{HSASM}(2m+1)$ is the set of horizontally symmetric ASMs of order $2m+1$, $\mu(A)$ is the statistic used in the preceding enumeration, and $f(A)$ is the position of the $1$ in the first row. Refined decomposition conjecture. There exists a family of polynomials $B_{2m}(z,\rho)$ in $\{z,\rho\}$ such that, for every $m\geq 1$,

$$A_{2m+1}(z,\rho)=\rho B_{2m+1}(z,1)B_{2m+2}(z,\rho),$$ $$A_{2m}(z,\rho)=\rho(\rho+1)B_{2m}(z,1)B_{2m+1}(z,\rho).$$

This refines the previously conjectured decomposition by tracking the position of the $1$ in the first row. The source presents it as a proposal; no proof or resolution is supplied in the given text.