Positive Bernstein expansion conjecture for Bernoulli polynomial densities

For each positive integer $n$, let $b_n(u)$ denote the factorially normalized Bernoulli polynomial of degree $n$, and consider the polynomial probability density $1-2^n b_n(u)$ on $[0,1)$. A Bernstein basis of degree $n$ consists of the polynomials $\binom{n}{k}u^k(1-u)^{n-k}$ for $0\leq k\leq n$. Positive Bernstein expansion conjecture. For every positive integer $n$, the density $1-2^n b_n(u)$ can be expanded in this Bernstein basis with positive coefficients. The conjecture concerns positivity of the Bernstein coefficients at degree $n$, complementing the known positive-coefficient expansion in degree $2n-1$ described earlier in the paper.