Adelic hypergeometric homology freeness conjecture

For positive integers $d$ and $N$, and $\lambda \in L(\mathbb{C})$, consider the smooth hypersurface $U_{N,\lambda}^{(d)} \subset \mathbb{A}^{d+1}$ defined by

$$\prod_{i=0}^d (1-x_i^N)=\lambda.$$

The group $\mu_N^{d+1}$ acts naturally on this hypersurface. Let $\Lambda:=\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}(1)^{d+1}]]$. Adelic hypergeometric homology freeness conjecture. The inverse limit

$$\varprojlim_N H_d(U_{N,\lambda}^{(d)},\widehat{\mathbb{Z}})$$

is a free module of rank $d+1$ over $\Lambda$. This is proposed as the homological structure needed to construct an adelic analogue of the generalized hypergeometric function ${}_{d+1}F_d$ with all lower parameters equal to $1$; no resolution is supplied in the source.