Integrality conjecture for normalized orthogonal group moments

Let $a\geq b\geq c$ be nonnegative integers, let $n$ denote the dimension parameter in the normalized orthogonal-group integral $J$, and let $J\begin{pmatrix}a&&\\ &b&\\ &&c\end{pmatrix}$ denote the corresponding joint moment. Then the integrality conjecture. The quantity

$$\left[\frac{c}{0}\right]_{n-1}\left[\frac{b+c}{0}\right]_nJ\begin{pmatrix}a&&\\ &b&\\ &&c\end{pmatrix}$$

is an integer. The conjecture arises from explicit formulas for joint moments of diagonal entries of orthogonal matrices; its resolution is not specified in the source.