Conjecture relating XXZ-chain component sums to totally symmetric alternating sign matrices

Let $S_N=S_N(x,\tau)$ denote the sum of the components of the relevant ground-state vector, and let $A_{\mathrm{TS}}(2N+1;t,\tau)$ be the generating function for totally symmetric alternating sign matrices of size $2N+1$. Define

$$t=\frac{1+x}{\left(1+x(x-\tau)\right)^{1/2}}.$$

Component-sum conjecture. We have

$$S_N(x,\tau)=\left(1+x(x-\tau)\right)^{n/2}A_{\mathrm{TS}}(2N+1;t,\tau).$$

The conjecture is motivated by computations of all totally symmetric alternating sign matrices for $m=0,\dots,9$ and of the corresponding generating functions; the supplied text gives no proof or resolution.