Normality conjecture for the binary sequence of binomial set partition values

Let $a(n)$ be the integer sequence from Theorem, and define its binary reduction by

$$\beta(n)=a(n)\bmod 2.$$

A binary sequence is normal if every finite binary string of length $\ell\geq 1$ occurs in it with limiting frequency $2^{-\ell}$. Normality conjecture. The sequence $\beta(n)$ is normal.