The uniform-distribution conjecture for bases with four zeros in their Pisano period

Let $\Phi_{\beta^k}(n)$ denote the relevant base-$\beta^k$ digit sequence associated with the Fibonacci concatenation, and let $\omega(\beta)$ be the number of zeros in one Pisano period of $\beta$.

Uniform-distribution conjecture. For every base $\beta$ with $\omega(\beta)=4$, there exists a $K$ such that, for all $k\geq K$, the digits in the sequences $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed.

The claim is motivated by computational and heuristic evidence, including the verified normality of the Fibonacci concatenation in base $5$, but remains open for the stated family of bases.