Maximal complexity conjecture for classical constants

Let $\alpha$ be one of the classical constants $\pi$, $e$, $\log 2$, $\zeta(3)$, or $\sqrt 2$. For an integral base $b\geq 2$, write $p(\alpha,b,m)$ for the number of distinct blocks of length $m$ occurring in the base-$b$ expansion of $\alpha$.

Maximal complexity conjecture. The complexity of $\alpha$ satisfies

$$p(\alpha,b,m)=b^m,$$

for every positive integer $m$ and every base $b\geq 2$.

This is the maximal possible subword complexity and would follow from normality of the relevant expansions. The conjecture is presented as widely believed, but no resolution is given here.