Dense induced cycles in Hamming hypercubes

Determine whether there exists a universal constant R such that every Hamming hypercube $Q_n$ with $n>1$ contains an R-dense induced cycle.

Background

This question is proposed as a first step toward establishing asymptotic hamiltonicity of the Hamming hypercube sequence. It is presented as a variant of the snake-in-the-box problem.

References

Is there a universal constant $R$, such that $Q_n$ has an $R$-dense induced cycle for every $n>1$?

— Asymptotic connectivity, hamiltonicity and toughness  (2610.09992 - Georgakopoulos, 7 Oct 2026) in Problem, Section 1, subsection “The main problems”