Ferromagnetic-string probability formula conjecture

Let $N=2n+1$ be odd, and let $a_1,\ldots,a_k$ denote the local observables defining the ferromagnetic-string correlations. Ferromagnetic-string probability conjecture. The correlations satisfy

$$\frac{\langle a_1a_2\cdots a_{k-1}\rangle}{\langle a_1a_2\cdots a_k\rangle} =\frac{(2k-2)!(2k-1)!(2n+k)!(n-k)!}{(k-1)!(3k-2)!(2n-k+1)!(n+k-1)!}.$$

The formula is attributed in the paper to earlier work and is used to derive a thermodynamic-limit expression, but the supplied text does not establish it for all admissible $n$ and $k$.