Simple normality of √2 in base 2

Determine whether the real number √2 is simply normal to base 2, meaning that the digits 0 and 1 each occur with asymptotic frequency 1/2 in its dyadic expansion.

Background

The paper studies Isaac’s Condition (TU), which was proposed as a route toward proving that square roots of nonsquare natural numbers are simply normal to base 2. The paper disproves Condition (TU), including its full-sequence version, so Isaac’s reduction cannot establish simple normality for √2. However, the paper does not resolve the underlying normality question for √2 itself, which remains an independent open problem.