Enumeration conjectures for odd-order HTSASMs and DSASMs

Let $n$ be a nonnegative integer. An HTSASM is a half-turn-symmetric alternating sign matrix, and a DSASM is a diagonally symmetric alternating sign matrix. Let $\delta(n,n-1)$ denote the partition associated with the corresponding irreducible representation of $\operatorname{GL}_{2n+1}$, and let $\dim \operatorname{GL}_{2n+1}(\delta(n,n-1))$ denote its dimension.

Enumeration conjectures for odd-order HTSASMs and DSASMs. The number of $(2n+1)\times(2n+1)$ HTSASMs is

$$\#\mathcal{A}^{\mathrm{HTS}}_{2n+1}=3^{-n^2}\left(\dim \operatorname{GL}_{2n+1}(\delta(n,n-1))\right)^2,$$

and the number of $(2n+1)\times(2n+1)$ DSASMs is

$$\#\mathcal{A}^{\mathrm{DS}}_{2n+1}=3^{-n(n-1)/2}\dim \operatorname{GL}_{2n+1}(\delta(n,n-1)).$$

The paper states that the enumeration problems for odd-order HTSASMs and DSASMs remain open and presents these as reformulations of conjectures from the cited literature.