The unit-circle-rootedness conjecture for prism-slice generating polynomials

For $\mathbf{c}=(c_1,\ldots,c_n)inmathbb{Z}_{>0}^n$ and $0\leq mleq n-1$, define the generating polynomial

$$p_{n,m,\mathbf{c}}(x)=\sum_{ell=0}^{\infty}W(ell,n,m+1,\mathbf{c})x^{ell}.$$

Unit-circle-rootedness conjecture. Every complex root of $p_{n,m,\mathbf{c}}(x)$ lies on the unit circle $|z|=1$. This conjecture concerns the generating polynomials arising from lattice points in slices of prisms; its resolution is not supplied here.