Closed-boundary rotor-model groundstate sum conjecture

For the rotor model with closed boundary conditions, let $\psi_0$ be its groundstate wavefunction and let $S(N)$ denote the sum of its elements. For odd size $N=2m-1$, normalize the smallest element to $A_{\rm V}(2m-1;3)$. For even size $N=2m$, normalize the wavefunction to the smallest integer such that all elements are integers. Closed-boundary rotor-model conjecture. For odd sizes, all elements are integers and

$$S(2m-1)=3^{(m-1)^2}N_8(2m).$$

For even sizes,

$$S(2m)=3^{2\theta_m}A_{\rm V}(2m+1),$$

where

$$\theta_m=\left\lfloor\frac{(m-1)(m+2)}{3}\right\rfloor,$$

and the listed values are $0,1,3,6,9$ for $m=1,\ldots,5$. This conjecture concerns integer groundstate components and their combinatorial sums; it was checked up to $N=10$, but no proof is supplied.