Polynomial-time algorithms beating the overlap-gap threshold for independent sets

In the Erdos-Renyi graph $G(n, d/n)$ with large fixed average degree $d$, the largest independent set has size about $2 (\ln d / d)n$, while greedy and known polynomial-time algorithms reach about $(\ln d / d)n$. Does any polynomial-time algorithm output, with high probability, an independent set of size $(1 + \epsilon)(\ln d / d)n$ for a fixed $\epsilon > 0$ and all sufficiently large $d$? The answer is yes (an algorithm with proof) or no (a proof of hardness for all polynomial-time algorithms under a stated assumption).

An explicit polynomial-time algorithm with a proven guarantee above (ln d / d) n, or a reduction proving hardness from a standard complexity assumption.