The spectral index conjecture for tensor powers

Let $G$ be a connected hypergraph, let $G^{m,s}$ denote its generalized power hypergraph with parameters $m$ and $s$, and let $c(G)$ denote the cyclic index of $G$. Spectral index conjecture. One has

$$c(G^{m,s})=s\cdot c(G).$$

The paper establishes this equality when $c(G)=1$; the conjecture asserts it in general for the generalized power hypergraph construction.