The normality conjecture for bases with four zeros in their Pisano period

Let $\omega(\beta)$ denote the number of zeros in one Pisano period of $\beta$, and let the Fibonacci concatenation be the infinite sequence formed by concatenating the Fibonacci numbers. For a nonnegative integer $y$, consider the base $2^y\beta$.

Normality conjecture for bases with four zeros. For every base $\beta$ with $\omega(\beta)=4$ and every nonnegative integer $y$, the Fibonacci concatenation is normal in base $2^y\beta$.

The source presents this as a conjectural extension of the proved result for bases of the form $5^x2^y$ and supports it with computational and heuristic evidence; it remains open.