Kahn's independent-set bound for regular graphs

Let $G$ be a $d$-regular graph with $n$ vertices, and let $i(G)$ denote its number of independent sets. Alon and Kahn's conjecture. One has

$$i(G) \leq i(K_{d,d})^{n/(2d)}=(2^{d+1}-1)^{n/(2d)}.$$

This formalized the proposed extremal role of a disjoint union of copies of $K_{d,d}$; the conjecture was subsequently resolved by Sah, who proved the relevant upper bound.