Alon–Kahn conjecture on independent sets in regular graphs

Let $G$ be a simple, finite, undirected graph with $N$ vertices that is $d$-regular, and let $i(G)$ denote its number of independent sets. The graph $\frac{N}{2d}K_{d,d}$ is the disjoint union of $\frac{N}{2d}$ copies of the complete bipartite graph $K_{d,d}$. Alon–Kahn conjecture. For every such graph $G$,

$$i(G) \leq \left(2^{d+1}-1\right)^\frac{N}{2d}.$$

Kahn proved this bound for regular bipartite graphs, while the conjecture asserts it for all regular graphs; the general case remains open in the supplied text.