Integrality conjecture for a truncated hypergeometric series

Let $d\geq 2$ be an integer. Let $n$ be a positive integer satisfying

$$n\equiv -1\pmod d,\qquad n>d-1.$$

Integrality conjecture. Then

$$\frac{(n-1)!^d d^{dn-d}}{n^2}\cdot {}_dF_{d-1}\bigg[\begin{matrix}\frac1d-1,&\frac1d,&1+\frac1d,&1+\frac1d,&\ldots,&1+\frac1d\\ &1,&1,&1,&\ldots,&1\end{matrix}\bigg|\,1\bigg]_{n-1}\in\mathbb Z.$$

This conjecture is motivated by a related congruence modulo $p^2$ and numerical calculations. The source gives no resolution, so the integrality assertion remains open.