Validity of the numerically motivated Sharpe Ratio bias bound for n > 10

Prove that the numerically motivated bound for the bias of the IID-resampled Sharpe Ratio for rolling-window mean-variance portfolios, bias(Θ_p^*) ≤ θ − (θ^2 + ψ) / sqrt(θ^2 + C ψ) with C = 3θ^2 + ψ + 1 (where θ is the risky asset’s Sharpe Ratio and ψ is the lag-1 autocorrelation of returns), holds for estimation window sizes n > 10 under the paper’s modeling assumptions.

Background

The paper derives analytical bounds for the bias of IID-resampled Sharpe Ratio estimates but finds them uninformative near θ ≈ 0. It therefore proposes an alternative numerically motivated bound and presents evidence that it appears to hold for practical window sizes.

Based on extensive experiments, the authors formulate a conjecture that this bound is valid for n > 10, but they do not provide a rigorous proof, making formal validation an open problem.