Optimal query exponent for Fourier-depth separations

Determine the optimal exponent governing the number of oracle queries for which the depth-k lower-level circuits remain unable to solve the Forrelation-based promise problems separating Fourier depths k and k+1, in both the phase-query and standard-query models.

Background

The paper establishes oracle separations between consecutive Fourier-hierarchy levels using Forrelation, but the quantitative lower bounds apply only up to query thresholds of the form N^{c_k}, where c_k is on the order of 1/k^2. The authors record a more general parameterized bound, but do not determine the optimal exponent. The unresolved issue concerns strengthening the query lower bound rather than obtaining the qualitative separation itself.