Typical-order upper bound for the deficit

Prove that for almost all primes p, the deficit Δ(p)=⌊(p/2)^{1/3}⌋−δ(p) satisfies Δ(p)≤p^{1/6+o(1)}, and consequently establish the conjectured typical behavior Δ(p)=p^{1/6+o(1)} for almost all primes.

Background

The paper proves a lower bound Δ(p)≥p^{1/6-o(1)} for almost all primes, contradicting an earlier conjecture that the gap is O(log p). It then develops a heuristic suggesting that the lower bound has the correct order of magnitude.

The missing part is an upper bound of order p^{1/6+o(1)} for almost all primes. The authors explain that the required assumptions about the number and distribution of viable cubic values are unproved, even under the Bateman–Horn conjecture, and formulate the resulting typical-deficit statement as a conjecture.