Ground-state transverse matrix-element formula conjecture

Let $N=2n+1$ be odd, let $|S_z=\pm1/2\rangle$ denote the two ground states, and define the transverse matrix element

$$M=\langle S_z=1/2\mid\sigma^x_j\mid S_z=-1/2\rangle.$$

Transverse matrix-element conjecture. The first-order splitting matrix element satisfies

$$(2n+1)M=\frac{4\cdot7\cdots(3n+1)}{2\cdot5\cdots(3n-1)}.$$

The formula agrees with the data reported in the paper, but no proof or general resolution is supplied.