Quadratic discrepancy conjecture for the Pref-opposite construction

A de Bruijn sequence of order $n$ is a cyclic binary sequence in which every binary word of length $n$ occurs exactly once; the Pref-opposite algorithm is the construction specified in the source. The discrepancy of a binary sequence is the maximum, over all prefixes, of the absolute difference between the numbers of $1$s and $0$s.

Quadratic discrepancy conjecture. The de Bruijn sequence constructed by the Pref-opposite algorithm has discrepancy $\Theta(n^2)$.

The conjecture is supported by computations for $10\leq n\leq 30$ and by an observed connection with OEIS sequence A033638, whose shifted formula gives the displayed experimental values; no proof or resolution is supplied here.