Ratio of independence number to uniform occupancy in triangle-free graphs

Prove that if G is triangle-free with minimum degree d, then α(G)/[α_G(1)|V(G)|] ≥ 2−o_d(1) as d tends to infinity.

Background

The paper compares the independence number α(G) with the average size of a uniformly random independent set, α_G(1)|V(G)|. It explains that improving the constant in the best-known upper bound for the Ramsey number R(3,k) would follow from a suitable separation between these quantities. The stated conjecture predicts an asymptotic factor of two for triangle-free graphs whose minimum degree tends to infinity and would imply R(3,k)≲k²/(2 log k).