Existence of a highly normal number that is not KL-stochastic

A real number $x$ is highly normal if it is weakly $f$-normal for every computable function $f$. A real number is KL-stochastic if it satisfies the corresponding Kolmogorov–Loveland stochasticity condition.

Existence conjecture. There is a highly normal number that is not KL-stochastic.

The paper has established that highly normal numbers are closed under complements and that some highly normal numbers are not supernormal. The separation between high normality and KL-stochasticity is left as an open problem.