Bernoulli polynomial approximation conjecture for the discrete differences

Let $b_n$ be the factorially normalized Bernoulli polynomial of degree $n$, and let $\delta_{k:2n}$ denote the difference between the uniform distribution on $\{1,\dots,2n\}$ and the distribution of the index $I_n$, for $1\leq k\leq 2n$. Thus, $\delta_{k:2n}=\frac{1}{2n}-p_{k:2n}$. Bernoulli polynomial approximation conjecture. The discrete sequence $(\delta_{1:2n},\dots,\delta_{2n:2n})$ satisfies

$$\sup_{1\leq k\leq 2n}\left|2n\pi^n\delta_{k:2n}-(2\pi)^n b_n\left(\frac{k-1}{2n-1}\right)\right|\xrightarrow[]{}0$$

as $n\to\infty$. This conjecture proposes that the discrete sequence approximates the Bernoulli polynomials, and consequently the associated shifted cosine functions, in the large-degree limit.