Maximum discrepancy conjecture for the de Bruijn sequence DB_max

Let $\mathcal{DB}_{\max}(n)$ denote the de Bruijn sequence of order $n$ defined in the source, and let the discrepancy of a binary sequence be the maximum, over all prefixes, of the absolute difference between the numbers of $1$s and $0$s.

Maximum discrepancy conjecture. The sequence $\mathcal{DB}_{\max}(n)$ has discrepancy equal to

$${n-1 \choose \lfloor n/2 \rfloor}+\lfloor n/2\rfloor,$$

and this is the maximum possible discrepancy over all de Bruijn sequences of order $n$.

The claim is motivated by an established lower bound and exact computations for $10\leq n\leq 25$; the source does not provide a proof of equality or of global maximality.