The C-series sum-of-entries conjecture

At $q=e^{2i\pi/3}$, let

be the ground-state eigenvector of the $C_r$ Hamiltonian in the homogeneous limit, normalized so that its smallest entry $_{_0}$ is $1$. Let $A(n)$ denote the number of Alternating Sign Matrices of size $n$. **C-series \sum conjecture.** The \sum of the entries of

is

$$\sum_\pi \Psi_\pi=A(\lfloor r/2\rfloor)A(\lceil r/2\rceil).$$

For odd $r=2n-1$, the preceding discussion presents the corresponding product as computed, while the even case follows by a specialization from the odd case; the displayed assertion is nevertheless placed in a conjecture environment in the source.